Problems 2301 to 2400

Problem 2301


ODE	\[ \boxed {y^{\prime \prime }=\tan \left (x \right ) \sec \left (x \right )} \] With initial conditions \begin {align} \left [y \left (0\right ) = \frac {\pi }{4}, y^{\prime }\left (0\right ) = 1\right ] \end {align}

program solution	\[ y = \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+\frac {\pi }{4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+\frac {\pi }{4} \]

Problem 2302


ODE	\[ \boxed {2 y^{\prime \prime }-{\mathrm e}^{y}=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = -2 \ln \left (1-\frac {x}{2}\right ) \] Veriﬁed OK. \[ y = -2 \ln \left (1+\frac {x}{2}\right ) \] Warning, solution could not be veriﬁed

Maple solution	\[ y \left (x \right ) = 2 \ln \left (2\right )+\ln \left (\frac {1}{\left (-2+x \right )^{2}}\right ) \]

Problem 2303


ODE	\[ \boxed {y^{\prime \prime }-y^{3}=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = \frac {\sqrt {2}}{2}\right ] \end {align}

program solution	\[ y = -\frac {\sqrt {2}}{x +\sqrt {2}} \] Veriﬁed OK. \[ y = -\frac {\sqrt {2}}{\sqrt {2}-x} \] Warning, solution could not be veriﬁed

Maple solution	\[ y \left (x \right ) = -\frac {\sqrt {2}}{x +\sqrt {2}} \]

Problem 2304


ODE	\[ \boxed {y^{\prime \prime }-{y^{\prime }}^{2} \cos \left (x \right )=0} \] With initial conditions \begin {align} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = -\frac {2}{\tan \left (\frac {x}{2}\right )-1} \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 2305


ODE	\[ \boxed {y y^{\prime \prime }-y^{2} y^{\prime }-{y^{\prime }}^{2}=0} \] With initial conditions \begin {align} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = -\frac {6}{{\mathrm e}^{\frac {3 x}{2}}-4} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\frac {6}{{\mathrm e}^{\frac {3 x}{2}}-4} \]

Problem 2306


ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}=-1} \] With initial conditions \begin {align} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align}

program solution	\[ y = -x +2 \ln \left (x +1\right )+1 \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -x +2 \ln \left (-x -1\right )-2 i \pi +1 \]

Problem 2307


ODE	\[ \boxed {y y^{\prime \prime }-y^{3}-{y^{\prime }}^{2}=0} \] With initial conditions \begin {align} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align}

program solution	\[ y = \tanh \left (\frac {\left (-i \sqrt {2}\, \pi +2 \sqrt {2}\, \operatorname {arccoth}\left (\sqrt {2}\right )-2 x \right ) \sqrt {2}}{4}\right )^{2}-1 \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\operatorname {sech}\left (\frac {\sqrt {2}\, \left (x -\sqrt {2}\, \operatorname {arctanh}\left (\sqrt {2}\right )\right )}{2}\right )^{2} \]

Problem 2308


ODE	\[ \boxed {\left ({y^{\prime }}^{2}+1\right )^{2}-y^{2} y^{\prime \prime }=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = \sqrt {2}\right ] \end {align}

program solution	\[ -\frac {\sqrt {-4 y^{2}+30 y-36}}{4}-\frac {9 \arcsin \left (\frac {4 y}{9}-\frac {5}{3}\right )}{8} = x -\frac {3 \sqrt {2}}{4}+\frac {9 \arcsin \left (\frac {1}{3}\right )}{8} \] Veriﬁed OK. \[ \frac {\sqrt {-4 y^{2}+30 y-36}}{4}+\frac {9 \arcsin \left (\frac {4 y}{9}-\frac {5}{3}\right )}{8} = x +\frac {3 \sqrt {2}}{4}-\frac {9 \arcsin \left (\frac {1}{3}\right )}{8} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \operatorname {RootOf}\left (\sqrt {2}\, \left (\int _{\textit {\_Z}}^{3}\frac {\operatorname {RootOf}\left (\left (-\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1}{\sqrt {-\left (\operatorname {RootOf}\left (\left (-\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1\right ) \left (2 \operatorname {RootOf}\left (\left (-\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} +\textit {\_a} -2\right )}}d \textit {\_a} \right )+x \right ) \\ y \left (x \right ) &= \operatorname {RootOf}\left (\sqrt {2}\, \left (\int _{3}^{\textit {\_Z}}\frac {\operatorname {RootOf}\left (\left (\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1}{\sqrt {-\left (\operatorname {RootOf}\left (\left (\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1\right ) \left (2 \operatorname {RootOf}\left (\left (\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} +\textit {\_a} -2\right )}}d \textit {\_a} \right )+x \right ) \\ \end{align}

Problem 2309


ODE	\[ \boxed {y^{\prime \prime }-{y^{\prime }}^{2} \sin \left (x \right )=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = {\frac {1}{2}}\right ] \end {align}

program solution	\[ y = \tan \left (\frac {x}{2}\right ) \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 2310


ODE	\[ \boxed {2 y y^{\prime \prime }-y^{3}-2 {y^{\prime }}^{2}=0} \] With initial conditions \begin {align} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 0] \end {align}

program solution	\[ y = -1 \] Warning, solution could not be veriﬁed

Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (2 \,\operatorname {arctanh}\left (\sqrt {\textit {\_Z} +1}\right )+x \right ) \]

Problem 2311


ODE	\[ \boxed {x^{\prime \prime }-k^{2} x=0} \] With initial conditions \begin {align} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = v_{0}] \end {align}

program solution	\[ x = \frac {v_{0} \left ({\mathrm e}^{\operatorname {csgn}\left (k \right ) k t}-{\mathrm e}^{-k t}\right )}{k \left (\operatorname {csgn}\left (k \right )+1\right )} \] Veriﬁed OK.

Maple solution	\[ x \left (t \right ) = -\frac {v_{0} \left ({\mathrm e}^{-k t}-{\mathrm e}^{k t}\right )}{2 k} \]

Problem 2312


ODE	\[ \boxed {y y^{\prime \prime }-2 {y^{\prime }}^{2}-y^{2}=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = \sqrt {3}\right ] \end {align}

program solution	\[ \arctan \left (\sqrt {4 y^{2}-1}\right ) = x +\frac {\pi }{3} \] Veriﬁed OK. \[ \arctan \left (\frac {1}{\sqrt {4 y^{2}-1}}\right ) = x +\frac {\pi }{6} \] Warning, solution could not be veriﬁed

Maple solution	\[ y \left (x \right ) = \frac {1}{-\sqrt {3}\, \sin \left (x \right )+\cos \left (x \right )} \]

Problem 2313


ODE	\[ \boxed {\left (1-{\mathrm e}^{x}\right ) y^{\prime \prime }-y^{\prime } {\mathrm e}^{x}=0} \] With initial conditions \begin {align} [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = 1] \end {align}

program solution	\[ y = -\left (-1+{\mathrm e}\right ) \left (x +\ln \left (-1+{\mathrm e}\right )-\ln \left ({\mathrm e}^{x}-1\right )-1\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -\left (\ln \left ({\mathrm e}^{x}\right )+\ln \left (-1+{\mathrm e}\right )-\ln \left ({\mathrm e}^{x}-1\right )-1\right ) \left (-1+{\mathrm e}\right ) \]

Problem 2314


ODE	\[ \boxed {4 y^{2}-{y^{\prime }}^{2} x^{2}=0} \]

program solution	\[ y = \frac {c_{2}}{x^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= c_{1} x^{2} \\ y \left (x \right ) &= \frac {c_{1}}{x^{2}} \\ \end{align}

Problem 2315


ODE	\[ \boxed {x y {y^{\prime }}^{2}+\left (x +y\right ) y^{\prime }=-1} \]

program solution	\[ y = \sqrt {-2 x -2 c_{2}} \] Veriﬁed OK. \[ y = -\sqrt {-2 x -2 c_{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\ln \left (x \right )+c_{1} \\ y \left (x \right ) &= \sqrt {-2 x +c_{1}} \\ y \left (x \right ) &= -\sqrt {-2 x +c_{1}} \\ \end{align}

Problem 2316


ODE	\[ \boxed {\left (2 y-x^{2}\right ) {y^{\prime }}^{2}-2 x^{2} y {y^{\prime }}^{2}=-1} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 2317


ODE	\[ \boxed {x \left (-1+{y^{\prime }}^{2}\right )-2 y y^{\prime }=0} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {c_{1}^{2} x^{2}-1}{2 c_{1}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \frac {-c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align}

Problem 2318


ODE	\[ \boxed {\left (1-y^{2}\right ) {y^{\prime }}^{2}=1} \]

program solution	\[ -\frac {y \sqrt {1-y^{2}}}{2}-\frac {\arcsin \left (y\right )}{2} = x +c_{1} \] Veriﬁed OK. \[ \frac {y \sqrt {1-y^{2}}}{2}+\frac {\arcsin \left (y\right )}{2} = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \sin \left (\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+\textit {\_Z} +2 c_{1} -2 x \right )\right ) \\ y \left (x \right ) &= \sin \left (\operatorname {RootOf}\left (-\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )-\textit {\_Z} +2 c_{1} -2 x \right )\right ) \\ \end{align}

Problem 2319


ODE	\[ \boxed {x y {y^{\prime }}^{2}+\left (y x -1\right ) y^{\prime }-y=0} \]

program solution

Maple solution	\[ \text {No solution found} \]

Problem 2320


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}+y^{\prime } x y=2 x^{2}} \]

program solution	\[ y = \sqrt {x^{2}+2 c_{2}} \] Veriﬁed OK. \[ y = -\sqrt {x^{2}+2 c_{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \sqrt {x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {x^{2}+c_{1}} \\ y \left (x \right ) &= \sqrt {-2 x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-2 x^{2}+c_{1}} \\ \end{align}

Problem 2321


ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}-2 y^{\prime } x y+2 y^{2}=x^{2}} \]

program solution	\[ y = x \] Veriﬁed OK. \[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = -x \] Veriﬁed OK. \[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -x \\ y \left (x \right ) &= x \\ y \left (x \right ) &= \sqrt {-2 \sqrt {2}\, c_{1} x -c_{1}^{2}-x^{2}} \\ y \left (x \right ) &= \sqrt {2 \sqrt {2}\, c_{1} x -c_{1}^{2}-x^{2}} \\ y \left (x \right ) &= -\sqrt {-2 \sqrt {2}\, c_{1} x -c_{1}^{2}-x^{2}} \\ y \left (x \right ) &= -\sqrt {2 \sqrt {2}\, c_{1} x -c_{1}^{2}-x^{2}} \\ \end{align}

Problem 2322


ODE	\[ \boxed {{y^{\prime }}^{3}+\left (x +y-2 y x \right ) {y^{\prime }}^{2}-2 y^{\prime } x y \left (x +y\right )=0} \]

program solution	\[ y = c_{3} {\mathrm e}^{x^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= {\mathrm e}^{x^{2}} c_{1} \\ y \left (x \right ) &= 1+{\mathrm e}^{-x} c_{1} -x \\ y \left (x \right ) &= c_{1} \\ \end{align}

Problem 2323


ODE	\[ \boxed {y {y^{\prime }}^{2}+\left (y^{2}-x^{3}-x y^{2}\right ) y^{\prime }-x y \left (x^{2}+y^{2}\right )=0} \]

program solution	\[ \frac {\left (1+x^{2}+y^{2}\right ) {\mathrm e}^{-x^{2}}}{2}-c_{1} = 0 \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= {\mathrm e}^{-x} c_{1} \\ y \left (x \right ) &= \sqrt {{\mathrm e}^{x^{2}} c_{1} -x^{2}-1} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{x^{2}} c_{1} -x^{2}-1} \\ \end{align}

Problem 2324


ODE	\[ \boxed {y-y^{\prime } x \left (y^{\prime }+1\right )=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ x = \frac {4 c_{2} x^{2} {\mathrm e}^{\frac {2 x}{-x +\sqrt {x \left (x +4 y\right )}}}}{\left (-x +\sqrt {x \left (x +4 y\right )}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {4 c_{2} x^{2} {\mathrm e}^{-\frac {2 x}{x +\sqrt {x \left (x +4 y\right )}}}}{\left (x +\sqrt {x \left (x +4 y\right )}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x \left (1+2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ y \left (x \right ) &= \frac {x \left (1+2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ \end{align}

Problem 2325


ODE	\[ \boxed {y-3 \ln \left (y^{\prime }\right )=x} \]

program solution	\[ y = x \] Veriﬁed OK. \[ y = x +3 \ln \left (-\frac {1}{{\mathrm e}^{\frac {x}{3}} c_{1} -1}\right ) \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= x \\ y \left (x \right ) &= x +3 \ln \left (\frac {{\mathrm e}^{-\frac {x}{3}} c_{1}}{-1+c_{1} {\mathrm e}^{-\frac {x}{3}}}\right ) \\ \end{align}

Problem 2326


ODE	\[ \boxed {y \left ({y^{\prime }}^{2}+1\right )=2} \]

program solution	\[ \frac {y \left (y-2\right )}{\sqrt {-y \left (y-2\right )}}+\arcsin \left (y-1\right ) = x +c_{1} \] Veriﬁed OK. \[ \sqrt {2 y-y^{2}}-\arcsin \left (y-1\right ) = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 2 \\ y \left (x \right ) &= -\sin \left (\operatorname {RootOf}\left (-\textit {\_Z} -x -\operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+c_{1} \right )\right )+1 \\ y \left (x \right ) &= \sin \left (\operatorname {RootOf}\left (-\textit {\_Z} -x +\operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+c_{1} \right )\right )+1 \\ \end{align}

Problem 2327


ODE	\[ \boxed {y {y^{\prime }}^{2}-2 y^{\prime } x +y=0} \]

program solution	\[ y = -x \] Veriﬁed OK. \[ y = 0 \] Veriﬁed OK. \[ y = x \] Veriﬁed OK. \[ x = \frac {2 c_{3} x}{x +\sqrt {x^{2}-y^{2}}} \] Veriﬁed OK. \[ x = -\frac {2 c_{3} x}{-x +\sqrt {x^{2}-y^{2}}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -x \\ y \left (x \right ) &= x \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (-2 i x +c_{1} \right )} \\ y \left (x \right ) &= \sqrt {c_{1} \left (2 i x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (-2 i x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (2 i x +c_{1} \right )} \\ \end{align}

Problem 2328


ODE	\[ \boxed {{y^{\prime }}^{2}+y^{2}=1} \]

program solution	\[ y = \sin \left (x +c_{1} \right ) \] Veriﬁed OK. \[ y = -\sin \left (x +c_{2} \right ) \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -1 \\ y \left (x \right ) &= 1 \\ y \left (x \right ) &= -\sin \left (c_{1} -x \right ) \\ y \left (x \right ) &= \sin \left (c_{1} -x \right ) \\ \end{align}

Problem 2329


ODE	\[ \boxed {x \left (-1+{y^{\prime }}^{2}\right )-2 y y^{\prime }=0} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {c_{1}^{2} x^{2}-1}{2 c_{1}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \frac {-c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align}

Problem 2330


ODE	\[ \boxed {-2 y y^{\prime }+{y^{\prime }}^{2} x=-4 x} \]

program solution	\[ y = -2 x \] Veriﬁed OK. \[ y = 2 x \] Veriﬁed OK. \[ y = \frac {c_{1}^{2} x^{2}+4}{2 c_{1}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -2 x \\ y \left (x \right ) &= 2 x \\ y \left (x \right ) &= \frac {4 c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align}

Problem 2331


ODE	\[ \boxed {2 x^{2} y+{y^{\prime }}^{2}-y^{\prime } x^{3}=0} \]

program solution	\[ \frac {\ln \left (y\right )}{4}-\frac {\ln \left (-x^{2}+\sqrt {x^{4}-8 y}\right )}{4}+\frac {\ln \left (x^{2}+\sqrt {x^{4}-8 y}\right )}{4} = c_{1} \] Veriﬁed OK. \[ \frac {\ln \left (y\right )}{4}+\frac {\ln \left (-x^{2}+\sqrt {x^{4}-8 y}\right )}{4}-\frac {\ln \left (x^{2}+\sqrt {x^{4}-8 y}\right )}{4} = c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x^{4}}{8} \\ y \left (x \right ) &= c_{1} \left (x^{2}-2 c_{1} \right ) \\ \end{align}

Problem 2332


ODE	\[ \boxed {y {y^{\prime }}^{2}-3 y^{\prime } x -y=0} \]

program solution	\[ y = -2 x \] Veriﬁed OK. \[ y = 0 \] Veriﬁed OK. \[ y = 2 x \] Veriﬁed OK. \[ y = \frac {3 {\operatorname {RootOf}\left (\left (c_{2}^{24}-x^{8}\right ) \textit {\_Z}^{64}+\left (-8 c_{2}^{24}+20 x^{8}\right ) \textit {\_Z}^{56}+\left (28 c_{2}^{24}-160 x^{8}\right ) \textit {\_Z}^{48}+\left (-56 c_{2}^{24}+640 x^{8}\right ) \textit {\_Z}^{40}+\left (70 c_{2}^{24}-1280 x^{8}\right ) \textit {\_Z}^{32}+\left (-56 c_{2}^{24}+1024 x^{8}\right ) \textit {\_Z}^{24}+28 c_{2}^{24} \textit {\_Z}^{16}-8 c_{2}^{24} \textit {\_Z}^{8}+c_{2}^{24}\right )}^{4} x}{{\operatorname {RootOf}\left (\left (c_{2}^{24}-x^{8}\right ) \textit {\_Z}^{64}+\left (-8 c_{2}^{24}+20 x^{8}\right ) \textit {\_Z}^{56}+\left (28 c_{2}^{24}-160 x^{8}\right ) \textit {\_Z}^{48}+\left (-56 c_{2}^{24}+640 x^{8}\right ) \textit {\_Z}^{40}+\left (70 c_{2}^{24}-1280 x^{8}\right ) \textit {\_Z}^{32}+\left (-56 c_{2}^{24}+1024 x^{8}\right ) \textit {\_Z}^{24}+28 c_{2}^{24} \textit {\_Z}^{16}-8 c_{2}^{24} \textit {\_Z}^{8}+c_{2}^{24}\right )}^{8}-1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= 0 \\ \ln \left (x \right )-\frac {3 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {\frac {9 x^{2}+4 y \left (x \right )^{2}}{x^{2}}}}\right )}{8}+\frac {5 \,\operatorname {arctanh}\left (\frac {9 x +8 y \left (x \right )}{5 x \sqrt {\frac {9 x^{2}+4 y \left (x \right )^{2}}{x^{2}}}}\right )}{16}-\frac {5 \,\operatorname {arctanh}\left (\frac {-9 x +8 y \left (x \right )}{5 x \sqrt {\frac {9 x^{2}+4 y \left (x \right )^{2}}{x^{2}}}}\right )}{16}+\frac {5 \ln \left (\frac {y \left (x \right )+2 x}{x}\right )}{16}+\frac {5 \ln \left (\frac {-2 x +y \left (x \right )}{x}\right )}{16}+\frac {3 \ln \left (\frac {y \left (x \right )}{x}\right )}{8}-c_{1} &= 0 \\ \ln \left (x \right )+\frac {3 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {\frac {9 x^{2}+4 y \left (x \right )^{2}}{x^{2}}}}\right )}{8}-\frac {5 \,\operatorname {arctanh}\left (\frac {9 x +8 y \left (x \right )}{5 x \sqrt {\frac {9 x^{2}+4 y \left (x \right )^{2}}{x^{2}}}}\right )}{16}+\frac {5 \,\operatorname {arctanh}\left (\frac {-9 x +8 y \left (x \right )}{5 x \sqrt {\frac {9 x^{2}+4 y \left (x \right )^{2}}{x^{2}}}}\right )}{16}+\frac {5 \ln \left (\frac {y \left (x \right )+2 x}{x}\right )}{16}+\frac {5 \ln \left (\frac {-2 x +y \left (x \right )}{x}\right )}{16}+\frac {3 \ln \left (\frac {y \left (x \right )}{x}\right )}{8}-c_{1} &= 0 \\ \end{align}

Problem 2333


ODE	\[ \boxed {-y {y^{\prime }}^{2}=-8 x -1} \]

program solution	\[ y = 2 x +\frac {1}{4} \] Veriﬁed OK. \[ y = \frac {8 x +1}{-2+2 i \sqrt {3}} \] Veriﬁed OK. \[ y = \frac {-8 x -1}{2 i \sqrt {3}+2} \] Veriﬁed OK. \[ x = -\frac {1}{8}+\frac {c_{3} \left (8 x +1\right )}{\left (\frac {\sqrt {y \left (8 x +1\right )}-2 y}{y}\right )^{\frac {2}{3}} \left (\frac {8 x +1+2 \sqrt {y \left (8 x +1\right )}+4 y}{y}\right )^{\frac {2}{3}} y} \] Veriﬁed OK. \[ x = -\frac {1}{8}+\frac {c_{3} \left (8 x +1\right )}{\left (\frac {-\sqrt {y \left (8 x +1\right )}-2 y}{y}\right )^{\frac {2}{3}} \left (\frac {8 x +1-2 \sqrt {y \left (8 x +1\right )}+4 y}{y}\right )^{\frac {2}{3}} y} \] Veriﬁed OK.

Maple solution	\begin{align} -\frac {8 c_{1} \left (8 x +1\right )}{\left (\frac {-2 y \left (x \right )-\sqrt {y \left (x \right ) \left (8 x +1\right )}}{y \left (x \right )}\right )^{\frac {2}{3}} \left (\frac {8 x +1-2 \sqrt {y \left (x \right ) \left (8 x +1\right )}+4 y \left (x \right )}{y \left (x \right )}\right )^{\frac {2}{3}} y \left (x \right )}+x +\frac {1}{8} &= 0 \\ \frac {1}{8}-\frac {8 c_{1} \left (8 x +1\right )}{\left (\frac {-2 y \left (x \right )+\sqrt {y \left (x \right ) \left (8 x +1\right )}}{y \left (x \right )}\right )^{\frac {2}{3}} \left (\frac {8 x +1+2 \sqrt {y \left (x \right ) \left (8 x +1\right )}+4 y \left (x \right )}{y \left (x \right )}\right )^{\frac {2}{3}} y \left (x \right )}+x &= 0 \\ \end{align}

Problem 2334


ODE	\[ \boxed {y {y^{\prime }}^{2}+2 y^{\prime }=-1} \]

program solution	\[ \frac {2 \left (1-y\right )^{\frac {3}{2}}}{3}+1-y = x +c_{1} \] Veriﬁed OK. \[ -\frac {2 \left (1-y\right )^{\frac {3}{2}}}{3}+1-y = x +c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} \frac {\left (2 y \left (x \right )-2\right ) \sqrt {1-y \left (x \right )}}{3}+x -c_{1} +y \left (x \right )-1 &= 0 \\ \frac {\left (-2 y \left (x \right )+2\right ) \sqrt {1-y \left (x \right )}}{3}+x -c_{1} +y \left (x \right )-1 &= 0 \\ \end{align}

Problem 2335


ODE	\[ \boxed {\left ({y^{\prime }}^{2}+1\right ) x -\left (x +y\right ) y^{\prime }=0} \]

program solution	\[ y = x \] Veriﬁed OK. \[ x = \frac {c_{2} \left (x +y+\sqrt {y^{2}+2 y x -3 x^{2}}\right ) {\mathrm e}^{\frac {x +y+\sqrt {y^{2}+2 y x -3 x^{2}}}{2 x}}}{2 x} \] Veriﬁed OK. \[ x = \frac {c_{2} \left (x +y-\sqrt {y^{2}+2 y x -3 x^{2}}\right ) {\mathrm e}^{\frac {x +y-\sqrt {y^{2}+2 y x -3 x^{2}}}{2 x}}}{2 x} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= x \\ y \left (x \right ) &= \frac {x \left (\operatorname {LambertW}\left (\frac {x}{c_{1}}\right )^{2}-\operatorname {LambertW}\left (\frac {x}{c_{1}}\right )+1\right )}{\operatorname {LambertW}\left (\frac {x}{c_{1}}\right )} \\ \end{align}

Problem 2336


ODE	\[ \boxed {-3 y y^{\prime }+{y^{\prime }}^{2} x=-x^{2}} \]

program solution	\[ y = \frac {\left (4 \,{\mathrm e}^{3 c_{1}}+x^{3}\right ) {\mathrm e}^{-\frac {3 c_{1}}{2}}}{6} \] Veriﬁed OK. \[ y = \frac {\left (4 x^{3}+{\mathrm e}^{3 c_{1}}\right ) {\mathrm e}^{-\frac {3 c_{1}}{2}}}{6} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {2 x^{\frac {3}{2}}}{3} \\ y \left (x \right ) &= \frac {2 x^{\frac {3}{2}}}{3} \\ y \left (x \right ) &= \frac {4 x^{3}+c_{1}^{2}}{6 c_{1}} \\ y \left (x \right ) &= \frac {c_{1}^{2} x^{3}+4}{6 c_{1}} \\ \end{align}

Problem 2337


ODE	\[ \boxed {y+2 y^{\prime } x -{y^{\prime }}^{2} x=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = 3 x \] Veriﬁed OK. \[ x = \frac {c_{3} x}{\left (-2 x +\sqrt {x \left (x +y\right )}\right ) \left (\frac {-2 x +\sqrt {x \left (x +y\right )}}{x}\right )^{\frac {1}{3}} \left (\frac {x +\sqrt {x \left (x +y\right )}}{x}\right )^{\frac {2}{3}}} \] Veriﬁed OK. \[ x = -\frac {c_{3} x}{\left (2 x +\sqrt {x \left (x +y\right )}\right ) \left (\frac {-2 x -\sqrt {x \left (x +y\right )}}{x}\right )^{\frac {1}{3}} \left (\frac {x -\sqrt {x \left (x +y\right )}}{x}\right )^{\frac {2}{3}}} \] Veriﬁed OK.

Maple solution	\begin{align} x \left (1-\frac {c_{1}}{\left (-2 x +\sqrt {x \left (x +y \left (x \right )\right )}\right ) \left (\frac {-2 x +\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{\frac {1}{3}} \left (\frac {x +\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{\frac {2}{3}}}\right ) &= 0 \\ x \left (1+\frac {c_{1}}{\left (2 x +\sqrt {x \left (x +y \left (x \right )\right )}\right ) \left (\frac {-2 x -\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{\frac {1}{3}} \left (\frac {x -\sqrt {x \left (x +y \left (x \right )\right )}}{x}\right )^{\frac {2}{3}}}\right ) &= 0 \\ \end{align}

Problem 2338


ODE	\[ \boxed {-{y^{\prime }}^{2}-y^{\prime }=-x} \]

program solution	\[ y = -\frac {x}{2}+\frac {\left (4 x +1\right )^{\frac {3}{2}}}{12}+c_{1} \] Veriﬁed OK. \[ y = -\frac {x}{2}-\frac {\left (4 x +1\right )^{\frac {3}{2}}}{12}+c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {\left (-4 x -1\right ) \sqrt {4 x +1}}{12}-\frac {x}{2}+c_{1} \\ y \left (x \right ) &= -\frac {x}{2}+\frac {\left (4 x +1\right )^{\frac {3}{2}}}{12}+c_{1} \\ \end{align}

Problem 2339


ODE	\[ \boxed {-y+{y^{\prime }}^{3}=-x} \]

program solution	\[ y = x +1 \] Veriﬁed OK. \[ x = \frac {3 \left (-x +y\right )^{\frac {2}{3}}}{2}+3 \left (-x +y\right )^{\frac {1}{3}}+3 \ln \left (\left (-x +y\right )^{\frac {1}{3}}-1\right )+c_{2} \] Veriﬁed OK. \[ x = -\frac {3 \left (-x +y\right )^{\frac {2}{3}}}{4}+\frac {3 i \sqrt {3}\, \left (-x +y\right )^{\frac {2}{3}}}{4}-\frac {3 \left (-x +y\right )^{\frac {1}{3}}}{2}-\frac {3 i \sqrt {3}\, \left (-x +y\right )^{\frac {1}{3}}}{2}-3 \ln \left (2\right )+3 \ln \left (-\left (-x +y\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-x +y\right )^{\frac {1}{3}}-2\right )+c_{2} \] Veriﬁed OK. \[ x = -\frac {3 \left (-x +y\right )^{\frac {2}{3}}}{4}-\frac {3 i \sqrt {3}\, \left (-x +y\right )^{\frac {2}{3}}}{4}-\frac {3 \left (-x +y\right )^{\frac {1}{3}}}{2}+\frac {3 i \sqrt {3}\, \left (-x +y\right )^{\frac {1}{3}}}{2}-3 \ln \left (2\right )+3 \ln \left (i \sqrt {3}\, \left (-x +y\right )^{\frac {1}{3}}-\left (-x +y\right )^{\frac {1}{3}}-2\right )+c_{2} \] Veriﬁed OK.

Maple solution	\begin{align} x -\frac {3 \left (-x +y \left (x \right )\right )^{\frac {2}{3}}}{2}-3 \left (-x +y \left (x \right )\right )^{\frac {1}{3}}-3 \ln \left (\left (-x +y \left (x \right )\right )^{\frac {1}{3}}-1\right )-c_{1} &= 0 \\ x +\frac {3 \left (-x +y \left (x \right )\right )^{\frac {2}{3}}}{4}-\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{\frac {2}{3}}}{4}+\frac {3 \left (-x +y \left (x \right )\right )^{\frac {1}{3}}}{2}+\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{\frac {1}{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (-4-2 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{\frac {1}{3}}-2 \left (-x +y \left (x \right )\right )^{\frac {1}{3}}\right )-c_{1} &= 0 \\ x +\frac {3 \left (-x +y \left (x \right )\right )^{\frac {2}{3}}}{4}+\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{\frac {2}{3}}}{4}+\frac {3 \left (-x +y \left (x \right )\right )^{\frac {1}{3}}}{2}-\frac {3 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{\frac {1}{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (2 i \sqrt {3}\, \left (-x +y \left (x \right )\right )^{\frac {1}{3}}-2 \left (-x +y \left (x \right )\right )^{\frac {1}{3}}-4\right )-c_{1} &= 0 \\ \end{align}

Problem 2340


ODE	\[ \boxed {2 y y^{\prime }-{y^{\prime }}^{2} x=-x} \]

program solution	\[ y = -i x \] Veriﬁed OK. \[ y = i x \] Veriﬁed OK. \[ y = \frac {c_{1}^{2} x^{2}-1}{2 c_{1}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \frac {-c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align}

Problem 2341


ODE	\[ \boxed {-2 y y^{\prime }+{y^{\prime }}^{2} x=-4 x} \]

program solution	\[ y = -2 x \] Veriﬁed OK. \[ y = 2 x \] Veriﬁed OK. \[ y = \frac {c_{1}^{2} x^{2}+4}{2 c_{1}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -2 x \\ y \left (x \right ) &= 2 x \\ y \left (x \right ) &= \frac {4 c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align}

Problem 2342


ODE	\[ \boxed {x {y^{\prime }}^{3}-y y^{\prime }=1} \]

program solution	\[ y = x -1 \] Veriﬁed OK. \[ x = -\frac {54 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} 3^{\frac {1}{3}} \left (-\frac {x 2^{\frac {1}{3}} \left (\sqrt {\frac {-4 y^{3}+27 x}{x}}\, c_{1} 3^{\frac {1}{6}}+2 \left (y+\frac {3 c_{1}}{2}\right ) 3^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{3}-\frac {2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2} \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}-3 x 3^{\frac {1}{3}} \left (\frac {2 c_{1} y^{2}}{9}+x \right ) 2^{\frac {2}{3}}+{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (-\frac {4 y c_{1}}{3}+x \right )\right ) 2^{\frac {2}{3}}}{\left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} y x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} x y+{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} \] Warning, solution could not be veriﬁed \[ x = -\frac {36 x^{3} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (\frac {8 y c_{1}}{9}-\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (c_{1} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (y+\frac {3 c_{1}}{2}\right ) \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (\frac {2 c_{1} y^{2}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 2^{\frac {2}{3}}}{{\left (\left (\sqrt {3}+i\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x y \left (i 3^{\frac {1}{3}}-3^{\frac {5}{6}}\right ) 2^{\frac {2}{3}}\right )}^{2} {\left (-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} y \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right )\right )}^{2}} \] Warning, solution could not be veriﬁed \[ x = \frac {36 x^{3} 3^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} \left (\left (-\frac {8 y c_{1}}{9}+\frac {2 x}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (-\frac {2^{\frac {1}{3}} \left (c_{1} \left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}+6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) \left (y+\frac {3 c_{1}}{2}\right )\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{9}+\left (\frac {x \left (i 3^{\frac {1}{3}}+\frac {3^{\frac {5}{6}}}{3}\right ) \sqrt {\frac {-4 y^{3}+27 x}{x}}}{3}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) \left (\frac {2 c_{1} y^{2}}{9}+x \right )\right ) 2^{\frac {2}{3}}\right )\right ) 2^{\frac {2}{3}}}{{\left (-\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} y\right )\right )}^{2} {\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+x \left (3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) y 2^{\frac {2}{3}}\right )}^{2}} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} \frac {12 \left (2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )+x \left (\frac {2^{\frac {1}{3}} \left (3^{\frac {1}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+3 \,3^{\frac {2}{3}}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}}{2}+2^{\frac {2}{3}} 3^{\frac {1}{3}} y \left (x \right )^{2}\right )\right ) x^{3} c_{1} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}}{\left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2} \left (2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}+2 x \left (y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}}-3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )\right )^{2}}+x -\frac {18 x^{4} \left (2^{\frac {2}{3}} 3^{\frac {5}{6}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}\, x +2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+9 \,3^{\frac {1}{3}} 2^{\frac {2}{3}} x -3 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{\left (2 y \left (x \right ) 3^{\frac {2}{3}} 2^{\frac {1}{3}} x +2^{\frac {2}{3}} 3^{\frac {1}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}-6 x {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}\right )^{2} \left (y \left (x \right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} &= 0 \\ -\frac {3 x^{3} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} c_{1} \left (\frac {8 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}} y \left (x \right )}{9}+x \left (2^{\frac {1}{3}} \left (\left (\frac {i 3^{\frac {2}{3}}}{9}-\frac {3^{\frac {1}{6}}}{9}\right ) \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}-\frac {2 y \left (x \right )^{2} 2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )}{9}\right )\right )}{2 {\left (\left (i-\sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} y \left (x \right ) x \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right )\right )}^{2} \left (-\frac {2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{6}+x \left (-2 {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) y \left (x \right ) 2^{\frac {1}{3}}\right )\right )^{2}}+x +\frac {216 x^{4} 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}} 3^{\frac {1}{3}} \left (-{\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+y \left (x \right ) \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+\frac {x \left (-i 3^{\frac {1}{3}}-\frac {3^{\frac {5}{6}}}{3}\right ) 2^{\frac {2}{3}} \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}}{2}+\frac {3 x \left (-i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}\right )}{{\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {2}{3}}+\left (-i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x 2^{\frac {2}{3}} y \left (x \right )\right )}^{2} {\left (\frac {\left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left ({\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right )}^{2} x^{4}\right )}^{\frac {1}{3}}}{2}+x \left (6 i {\left (\left (\sqrt {3}\, \sqrt {\frac {-4 y \left (x \right )^{3}+27 x}{x}}+9\right ) x^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {2}{3}}+3 \,3^{\frac {1}{6}}\right ) y \left (x \right )\right )\right )}^{2}} &= 0 \\ \text {Expression too large to display} \\ \end{align}

Problem 2343


ODE	\[ \boxed {y \left ({y^{\prime }}^{2}+1\right )-2 y^{\prime } x=0} \]

program solution	\[ y = -x \] Veriﬁed OK. \[ y = 0 \] Veriﬁed OK. \[ y = x \] Veriﬁed OK. \[ x = \frac {2 c_{3} x}{x +\sqrt {x^{2}-y^{2}}} \] Veriﬁed OK. \[ x = -\frac {2 c_{3} x}{-x +\sqrt {x^{2}-y^{2}}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -x \\ y \left (x \right ) &= x \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (-2 i x +c_{1} \right )} \\ y \left (x \right ) &= \sqrt {c_{1} \left (2 i x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (-2 i x +c_{1} \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (2 i x +c_{1} \right )} \\ \end{align}

Problem 2344


ODE	\[ \boxed {-2 y y^{\prime }+{y^{\prime }}^{2} x=-2 x} \]

program solution	\[ y = \sqrt {2}\, x \] Veriﬁed OK. \[ y = -\sqrt {2}\, x \] Veriﬁed OK. \[ y = \frac {c_{1}^{2} x^{2}+2}{2 c_{1}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \sqrt {2}\, x \\ y \left (x \right ) &= -\sqrt {2}\, x \\ y \left (x \right ) &= \frac {2 c_{1}^{2}+x^{2}}{2 c_{1}} \\ \end{align}

Problem 2345


ODE	\[ \boxed {-y y^{\prime }-{y^{\prime }}^{2}=-x} \]

program solution	\[ y = 1-x \] Veriﬁed OK. \[ y = x -1 \] Veriﬁed OK. \[ x = \left (-y+\sqrt {y^{2}+4 x}\right ) \left (-\frac {-\ln \left (2\right )+\ln \left (-y+\sqrt {y^{2}+4 x}+\sqrt {2 y^{2}+4 x -2 y \sqrt {y^{2}+4 x}-4}\right )}{\sqrt {2 y^{2}+4 x -2 y \sqrt {y^{2}+4 x}-4}}+\frac {2 c_{1}}{\sqrt {-2 y+2 \sqrt {y^{2}+4 x}+4}\, \sqrt {-2 y+2 \sqrt {y^{2}+4 x}-4}}\right ) \] Veriﬁed OK. \[ x = \left (y+\sqrt {y^{2}+4 x}\right ) \left (\frac {-\ln \left (2\right )+\ln \left (-y-\sqrt {y^{2}+4 x}+\sqrt {2 y^{2}+4 x +2 y \sqrt {y^{2}+4 x}-4}\right )}{\sqrt {2 y^{2}+4 x +2 y \sqrt {y^{2}+4 x}-4}}-\frac {2 c_{1}}{\sqrt {-2 y-2 \sqrt {y^{2}+4 x}+4}\, \sqrt {-2 y-2 \sqrt {y^{2}+4 x}-4}}\right ) \] Veriﬁed OK.

Maple solution	\begin{align} \frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) c_{1}}{\sqrt {-2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}+4 x}+4}\, \sqrt {-2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}+4 x}-4}}+x +\frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) \left (-\ln \left (2\right )+\ln \left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}+\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}\right )\right )}{\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}} &= 0 \\ \frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) c_{1}}{\sqrt {-2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}+4 x}+4}\, \sqrt {-2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}+4 x}-4}}+x -\frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}+4 x}\right ) \left (-\ln \left (2\right )+\ln \left (-y \left (x \right )-\sqrt {y \left (x \right )^{2}+4 x}+\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}\right )\right )}{\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}+4 x}+4 x -4}} &= 0 \\ \end{align}

Problem 2346


ODE	\[ \boxed {4 {y^{\prime }}^{2} x +2 y^{\prime } x -y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ y = -\frac {x}{4} \] Veriﬁed OK. \[ y = \left (\frac {4 c_{1}^{2}}{x}+\frac {2 c_{1}}{\sqrt {x}}\right ) x \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {x}{4} \\ y \left (x \right ) &= 4 c_{1} +2 \sqrt {c_{1} x} \\ y \left (x \right ) &= 4 c_{1} -2 \sqrt {c_{1} x} \\ \end{align}

Problem 2347


ODE	\[ \boxed {y-y^{\prime } x \left (y^{\prime }+1\right )=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ x = \frac {4 c_{2} x^{2} {\mathrm e}^{\frac {2 x}{-x +\sqrt {x \left (x +4 y\right )}}}}{\left (-x +\sqrt {x \left (x +4 y\right )}\right )^{2}} \] Veriﬁed OK. \[ x = \frac {4 c_{2} x^{2} {\mathrm e}^{-\frac {2 x}{x +\sqrt {x \left (x +4 y\right )}}}}{\left (x +\sqrt {x \left (x +4 y\right )}\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {x \left (1+2 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (-\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ y \left (x \right ) &= \frac {x \left (1+2 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )\right )}{4 \operatorname {LambertW}\left (\frac {1}{2 \sqrt {\frac {c_{1}}{x}}}\right )^{2}} \\ \end{align}

Problem 2348


ODE	\[ \boxed {2 {y^{\prime }}^{3} x -{y^{\prime }}^{2} y=-1} \]

program solution	\[ y = \infty \] Veriﬁed OK. \[ x = \frac {36 \left (\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}} c_{1} +\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}} c_{1} y+c_{1} y^{2}-12 x \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}\right ) x^{2} \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}}{{\left (\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+y \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+y^{2}\right )}^{3}} \] Veriﬁed OK. \[ x = \frac {144 x^{2} \left (\left (-i \sqrt {3}\, c_{1} +c_{1} \right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+\left (-2 c_{1} y+24 x \right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) c_{1} y^{2}\right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}}{{\left (-i \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+i \sqrt {3}\, y^{2}+\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}-2 y \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+y^{2}\right )}^{3}} \] Veriﬁed OK. \[ x = \frac {144 x^{2} \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}} \left (c_{1} \left (-i \sqrt {3}-1\right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+2 \left (c_{1} y-12 x \right ) \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) c_{1} y^{2}\right )}{{\left (-i \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+i \sqrt {3}\, y^{2}-\left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {2}{3}}+2 y \left (6 \sqrt {3}\, \sqrt {-y^{3}+27 x^{2}}\, x +y^{3}-54 x^{2}\right )^{\frac {1}{3}}-y^{2}\right )}^{3}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {9 \left (-\frac {2 \,3^{\frac {2}{3}} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}} c_{1}^{2}}{9}+\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) \left (-\frac {2 \,3^{\frac {1}{3}} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} c_{1}}{9}+x \right ) x \right ) 3^{\frac {1}{3}} x^{2}}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} \left (c_{1} 3^{\frac {1}{3}} x +{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}\right )^{2}} \\ y \left (x \right ) &= \frac {4 \left (3 \left (-i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) c_{1}^{2} {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+\left (c_{1} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}+9 x \right ) x \left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right )\right ) 3^{\frac {1}{3}} x^{2}}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} {\left (\left (i-\sqrt {3}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+c_{1} x \left (i 3^{\frac {1}{3}}+3^{\frac {5}{6}}\right )\right )}^{2}} \\ y \left (x \right ) &= -\frac {4 \,3^{\frac {1}{3}} \left (-3 c_{1}^{2} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+\left (c_{1} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}}-9 x \right ) x \left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right )\right ) x^{2}}{{\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} {\left (\left (\sqrt {3}+i\right ) {\left (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+c_{1} \left (-3^{\frac {5}{6}}+i 3^{\frac {1}{3}}\right ) x \right )}^{2}} \\ \end{align}

Problem 2349


ODE	\[ \boxed {{y^{\prime }}^{3}+y^{\prime } x y-2 y^{2}=0} \]

program solution	\[ 3 \ln \left (x \right ) = \int _{}^{\frac {y}{x^{3}}}-\frac {9 \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}+\sqrt {27 \textit {\_a} +1}\right )^{\frac {1}{3}} 3^{\frac {1}{6}}}{\sqrt {\textit {\_a}}\, \left (-3^{\frac {2}{3}} \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}+\sqrt {27 \textit {\_a} +1}\right )^{\frac {2}{3}}+9 \,3^{\frac {1}{6}} \sqrt {\textit {\_a}}\, \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}+\sqrt {27 \textit {\_a} +1}\right )^{\frac {1}{3}}+3^{\frac {2}{3}}\right )}d \textit {\_a} +c_{1} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {x^{3}}{27} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (c_{1} x +1\right )^{2}}{4 c_{1}^{3}} \\ \end{align}

Problem 2350


ODE	\[ \boxed {3 {y^{\prime }}^{4} x -{y^{\prime }}^{3} y=1} \]

program solution	\[ x = \frac {5 \operatorname {RootOf}\left (3 x \,\textit {\_Z}^{4}-y \textit {\_Z}^{3}-1\right )^{3} c_{1} y+9 \operatorname {RootOf}\left (3 x \,\textit {\_Z}^{4}-y \textit {\_Z}^{3}-1\right )^{\frac {3}{2}} x +5 c_{1}}{15 x \operatorname {RootOf}\left (3 x \,\textit {\_Z}^{4}-y \textit {\_Z}^{3}-1\right )^{\frac {11}{2}}} \] Veriﬁed OK.

Maple solution	\[ \left [x \left (\textit {\_T} \right ) = \frac {5 c_{1} \textit {\_T}^{\frac {5}{2}}+3}{5 \textit {\_T}^{4}}, y \left (\textit {\_T} \right ) = \frac {15 c_{1} \textit {\_T}^{\frac {5}{2}}+4}{5 \textit {\_T}^{3}}\right ] \]

Problem 2351


ODE	\[ \boxed {2 {y^{\prime }}^{5}+2 y^{\prime } x -y=0} \]

program solution	\[ y = 0 \] Veriﬁed OK. \[ x = \frac {10 x \operatorname {RootOf}\left (2 \textit {\_Z}^{5}+2 x \textit {\_Z} -y\right )^{2}-5 y \operatorname {RootOf}\left (2 \textit {\_Z}^{5}+2 x \textit {\_Z} -y\right )+6 c_{1}}{6 \operatorname {RootOf}\left (2 \textit {\_Z}^{5}+2 x \textit {\_Z} -y\right )^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {20 \sqrt {5}\, \sqrt {-\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}\, \left (-\frac {3 \left (i \sqrt {3}-1\right ) \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (1+i \sqrt {3}\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}-90 c_{1} -6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {20 \sqrt {5}\, \sqrt {-\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 i \sqrt {3}\, x +\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}\, \left (-\frac {3 \left (i \sqrt {3}-1\right ) \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (1+i \sqrt {3}\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}-90 c_{1} -6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right )}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {20 \sqrt {5}\, \left (-\frac {3 \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (1+i \sqrt {3}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (i \sqrt {3}-1\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+90 c_{1} +6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right ) \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= \frac {20 \sqrt {5}\, \left (-\frac {3 \left (c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{15}\right ) \left (1+i \sqrt {3}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (\left (i \sqrt {3}-1\right ) x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+90 c_{1} +6 \sqrt {20 x^{3}+225 c_{1}^{2}}\right ) x \right ) \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (i \sqrt {3}\, \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 i \sqrt {3}\, x -\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}+20 x \right )}}{\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15000 c_{1} +1000 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= -\frac {\left (\frac {\left (3 c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{5}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+x \left (x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+45 c_{1} +3 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right ) \sqrt {10}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{25 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15 c_{1} +\sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ y \left (x \right ) &= \frac {\left (\frac {\left (3 c_{1} +\frac {\sqrt {20 x^{3}+225 c_{1}^{2}}}{5}\right ) \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+x \left (x \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}}+45 c_{1} +3 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )\right ) \sqrt {10}\, \sqrt {\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (\left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {2}{3}}-20 x \right )}}{25 \left (300 c_{1} +20 \sqrt {20 x^{3}+225 c_{1}^{2}}\right )^{\frac {1}{3}} \left (15 c_{1} +\sqrt {20 x^{3}+225 c_{1}^{2}}\right )} \\ \end{align}

Problem 2352


ODE	\[ \boxed {\frac {1}{{y^{\prime }}^{2}}+y^{\prime } x -2 y=0} \]

program solution	\[ y = \infty \] Warning, solution could not be veriﬁed \[ x = \frac {108 x^{3}}{\left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )^{3}}+\frac {c_{1} \left (\frac {16 y^{2}}{\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}}+4 y+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}\right )}{6 x} \] Veriﬁed OK. \[ x = \frac {-10368 x^{4} \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}-55296 y^{3} x^{3}+93312 x^{5}}{{\left (-i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}+\frac {c_{1} \left (\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}+8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 \left (1+i \sqrt {3}\right ) y^{2}\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \] Warning, solution could not be veriﬁed \[ x = \frac {3456 \left (-3 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x -16 y^{3}+27 x^{2}\right ) x^{3}}{{\left (i \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 i \sqrt {3}\, y^{2}+\left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y^{2}\right )}^{3}}-\frac {c_{1} \left (\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {2}{3}}-8 y \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}}-16 y^{2} \left (i \sqrt {3}-1\right )\right )}{12 \left (12 \sqrt {3}\, \sqrt {-32 y^{3}+27 x^{2}}\, x +64 y^{3}-108 x^{2}\right )^{\frac {1}{3}} x} \] Warning, solution could not be veriﬁed

Maple solution	\begin{align} \frac {279936 \left (-\frac {x \left (-\frac {4 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{108}-\frac {40 y \left (x \right )^{3} c_{1} x^{2}}{81}-\frac {x^{4} y \left (x \right )^{2}}{27}+\frac {c_{1} x^{4}}{12}+\frac {32 y \left (x \right )^{6} c_{1}}{81}+\frac {8 x^{2} y \left (x \right )^{5}}{243}\right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+34992 \left (-\frac {16 y \left (x \right )^{3}}{27}+x^{2}-\frac {x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{9}\right ) \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {64 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {32 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+279936 y \left (x \right ) \left (-\frac {\left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {128 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {40 x^{2} c_{1} y \left (x \right )}{3}\right ) x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}}{9}-\frac {1792 y \left (x \right )^{4} c_{1} x^{2}}{81}+\frac {40 y \left (x \right ) c_{1} x^{4}}{3}-\frac {16 x^{4} y \left (x \right )^{3}}{9}+\frac {128 y \left (x \right )^{6} x^{2}}{243}+x^{6}+\frac {512 y \left (x \right )^{7} c_{1}}{81}\right )}{{\left (\left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+4 y \left (x \right ) \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+16 y \left (x \right )^{2}\right )}^{3} x \left (12 x \sqrt {-96 y \left (x \right )^{3}+81 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}} &= 0 \\ \frac {1119744 \left (\frac {\left (i-\frac {\sqrt {3}}{3}\right ) x \left (-\frac {16 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{144}-\frac {\left (i \sqrt {3}-1\right ) \left (\frac {128 x^{2} y \left (x \right )^{5}}{81}+\frac {128 y \left (x \right )^{6} c_{1}}{27}-\frac {16 x^{4} y \left (x \right )^{2}}{9}-\frac {160 y \left (x \right )^{3} c_{1} x^{2}}{27}+c_{1} x^{4}\right )}{48}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}-279936 \left (-\frac {x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{9}-\frac {16 y \left (x \right )^{3}}{27}+x^{2}\right ) \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {16 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {8 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+1119744 y \left (x \right ) \left (-\frac {\left (i+\frac {\sqrt {3}}{3}\right ) \left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {32 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {10 x^{2} c_{1} y \left (x \right )}{3}\right ) x \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{3}+\left (\frac {128 y \left (x \right )^{6} x^{2}}{243}+\frac {128 y \left (x \right )^{7} c_{1}}{81}-\frac {16 x^{4} y \left (x \right )^{3}}{9}-\frac {448 y \left (x \right )^{4} c_{1} x^{2}}{81}+x^{6}+\frac {10 y \left (x \right ) c_{1} x^{4}}{3}\right ) \left (1+i \sqrt {3}\right )\right )}{\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}} \left (16 i \sqrt {3}\, y \left (x \right )^{2}-i \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}+16 y \left (x \right )^{2}-8 y \left (x \right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}\right )^{3} x} &= 0 \\ \frac {1119744 \left (\frac {\left (i+\frac {\sqrt {3}}{3}\right ) x \left (-\frac {16 x^{2} y \left (x \right )^{2}}{9}-\frac {16 y \left (x \right )^{3} c_{1}}{3}+c_{1} x^{2}\right ) \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{144}-\frac {\left (\frac {128 x^{2} y \left (x \right )^{5}}{81}+\frac {128 y \left (x \right )^{6} c_{1}}{27}-\frac {16 x^{4} y \left (x \right )^{2}}{9}-\frac {160 y \left (x \right )^{3} c_{1} x^{2}}{27}+c_{1} x^{4}\right ) \left (1+i \sqrt {3}\right )}{48}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}+279936 \left (-\frac {x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{9}-\frac {16 y \left (x \right )^{3}}{27}+x^{2}\right ) \left (-\frac {16 x^{2} y \left (x \right )^{3}}{9}-\frac {16 c_{1} y \left (x \right )^{4}}{3}+x^{4}+\frac {8 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}+1119744 \left (-\frac {\left (-\frac {32 x^{2} y \left (x \right )^{3}}{27}-\frac {32 c_{1} y \left (x \right )^{4}}{9}+x^{4}+\frac {10 x^{2} c_{1} y \left (x \right )}{3}\right ) \left (i-\frac {\sqrt {3}}{3}\right ) x \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}}{3}+\left (\frac {128 y \left (x \right )^{6} x^{2}}{243}+\frac {128 y \left (x \right )^{7} c_{1}}{81}-\frac {16 x^{4} y \left (x \right )^{3}}{9}-\frac {448 y \left (x \right )^{4} c_{1} x^{2}}{81}+x^{6}+\frac {10 y \left (x \right ) c_{1} x^{4}}{3}\right ) \left (i \sqrt {3}-1\right )\right ) y \left (x \right )}{\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}} \left (16 i \sqrt {3}\, y \left (x \right )^{2}-i \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}} \sqrt {3}-16 y \left (x \right )^{2}+8 y \left (x \right ) \left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {1}{3}}-\left (12 x \sqrt {3}\, \sqrt {-32 y \left (x \right )^{3}+27 x^{2}}+64 y \left (x \right )^{3}-108 x^{2}\right )^{\frac {2}{3}}\right )^{3} x} &= 0 \\ \end{align}

Problem 2353


ODE	\[ \boxed {2 y-3 y^{\prime } x -2 \ln \left (y^{\prime }\right )=4} \]

program solution	\[ y = -\infty \] Warning, solution could not be veriﬁed \[ x = -\frac {3 x \left (-9 c_{1} x^{2}+4 \operatorname {LambertW}\left (\frac {3 x \,{\mathrm e}^{y-2}}{2}\right )^{2}\right )}{8 \operatorname {LambertW}\left (\frac {3 x \,{\mathrm e}^{y-2}}{2}\right )^{3}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {\ln \left (\frac {\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}-2 \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+4}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\left (-\ln \left (2\right )-\ln \left (3\right )+\frac {3}{2}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\frac {\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}}{4}+1}{\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {-8 \ln \left (-\frac {\left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right ) \left (2+i \left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-2\right ) \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}\right )}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}+\left (16 \ln \left (2\right )+8 \ln \left (3\right )-12\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4 i \sqrt {3}+4}{8 \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {8 \ln \left (\frac {\left (-2+i \left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-2\right ) \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}\right ) \left (\left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right )}{x \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}}\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {2}{3}}+\left (-16 \ln \left (2\right )-8 \ln \left (3\right )+12\right ) \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4 i \sqrt {3}-4}{8 \left (12 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-4 c_{1}}\, x +108 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ \end{align}

Problem 2354


ODE	\[ \boxed {y-y^{\prime } x -{y^{\prime }}^{2}=0} \]

program solution	\[ y = c_{1}^{2}+c_{1} x \] Veriﬁed OK. \[ y = -\frac {x^{2}}{4} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {x^{2}}{4} \\ y \left (x \right ) &= c_{1} \left (c_{1} +x \right ) \\ \end{align}

Problem 2355


ODE	\[ \boxed {y-y^{\prime } x -\frac {1}{y^{\prime }}=0} \]

program solution	\[ y = c_{1} x +\frac {1}{c_{1}} \] Veriﬁed OK. \[ y = 2 \sqrt {x} \] Veriﬁed OK. \[ y = -2 \sqrt {x} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -2 \sqrt {x} \\ y \left (x \right ) &= 2 \sqrt {x} \\ y \left (x \right ) &= c_{1} x +\frac {1}{c_{1}} \\ \end{align}

Problem 2356


ODE	\[ \boxed {y-y^{\prime } x +\sqrt {y^{\prime }}=0} \]

program solution	\[ y = c_{1} x -\sqrt {c_{1}} \] Veriﬁed OK. \[ y = -\frac {\sqrt {\frac {1}{x^{2}}}}{4} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {1}{4 x} \\ y \left (x \right ) &= c_{1} x -\sqrt {c_{1}} \\ \end{align}

Problem 2357


ODE	\[ \boxed {y-y^{\prime } x -\ln \left (y^{\prime }\right )=0} \]

program solution	\[ y = c_{1} x +\ln \left (c_{1} \right ) \] Veriﬁed OK. \[ y = \ln \left (-\frac {1}{x}\right )-1 \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \ln \left (-\frac {1}{x}\right )-1 \\ y \left (x \right ) &= c_{1} x +\ln \left (c_{1} \right ) \\ \end{align}

Problem 2358


ODE	\[ \boxed {y-y^{\prime } x -\frac {3}{{y^{\prime }}^{2}}=0} \]

program solution	\[ y = c_{1} x +\frac {3}{c_{1}^{2}} \] Veriﬁed OK. \[ y = \frac {3 x^{2} 6^{\frac {1}{3}}}{2 \left (x^{2}\right )^{\frac {2}{3}}} \] Veriﬁed OK. \[ y = -\frac {9 x^{2} 2^{\frac {1}{3}}}{\left (x^{2}\right )^{\frac {2}{3}} \left (3 i 3^{\frac {1}{6}}+3^{\frac {2}{3}}\right )} \] Veriﬁed OK. \[ y = \frac {9 x^{2} 2^{\frac {1}{3}}}{\left (x^{2}\right )^{\frac {2}{3}} \left (3 i 3^{\frac {1}{6}}-3^{\frac {2}{3}}\right )} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \frac {3 \,6^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {3 \,2^{\frac {1}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) \left (x^{2}\right )^{\frac {1}{3}}}{4} \\ y \left (x \right ) &= \frac {3 \left (x^{2}\right )^{\frac {1}{3}} 2^{\frac {1}{3}} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )}{4} \\ y \left (x \right ) &= c_{1} x +\frac {3}{c_{1}^{2}} \\ \end{align}

Problem 2359


ODE	\[ \boxed {y-y^{\prime } x +{y^{\prime }}^{\frac {2}{3}}=0} \]

program solution	\[ y = c_{1} x -c_{1}^{\frac {2}{3}} \] Veriﬁed OK. \[ y = \frac {-\frac {4 \left (\frac {1}{x^{3}}\right )^{\frac {2}{3}} x^{2}}{9}+\frac {8}{27}}{x^{2}} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -\frac {4}{27 x^{2}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} x -c_{1}^{\frac {2}{3}} \\ \end{align}

Problem 2360


ODE	\[ \boxed {y-y^{\prime } x -{\mathrm e}^{y^{\prime }}=0} \]

program solution	\[ y = c_{1} x +{\mathrm e}^{c_{1}} \] Veriﬁed OK. \[ y = x \left (\ln \left (-x \right )-1\right ) \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= x \left (\ln \left (-x \right )-1\right ) \\ y \left (x \right ) &= c_{1} x +{\mathrm e}^{c_{1}} \\ \end{align}

Problem 2361


ODE	\[ \boxed {\left (y-y^{\prime } x \right )^{2}-{y^{\prime }}^{2}=1} \]

program solution	\[ y = c_{1} x +\sqrt {c_{1}^{2}+1} \] Veriﬁed OK. \[ y = \left (-x^{2}+1\right ) \sqrt {-\frac {1}{x^{2}-1}} \] Veriﬁed OK. \[ y = c_{2} x -\sqrt {c_{2}^{2}+1} \] Veriﬁed OK. \[ y = \sqrt {-\frac {1}{x^{2}-1}}\, \left (x^{2}-1\right ) \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= \sqrt {-x^{2}+1} \\ y \left (x \right ) &= -\sqrt {-x^{2}+1} \\ y \left (x \right ) &= c_{1} x -\sqrt {c_{1}^{2}+1} \\ y \left (x \right ) &= c_{1} x +\sqrt {c_{1}^{2}+1} \\ \end{align}

Problem 2362


ODE	\[ \boxed {{y^{\prime }}^{2} x -y y^{\prime }=2} \]

program solution	\[ y = c_{1} x -\frac {2}{c_{1}} \] Veriﬁed OK. \[ y = 2 \sqrt {2}\, \sqrt {-x} \] Veriﬁed OK. \[ y = -2 \sqrt {2}\, \sqrt {-x} \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= -2 \sqrt {2}\, \sqrt {-x} \\ y \left (x \right ) &= 2 \sqrt {2}\, \sqrt {-x} \\ y \left (x \right ) &= c_{1} x -\frac {2}{c_{1}} \\ \end{align}

Problem 2363


ODE	\[ \boxed {y^{2}-2 y^{\prime } x y+{y^{\prime }}^{2} \left (x^{2}-1\right )=0} \]

program solution	\[ y = c_{2} \left (x +1\right ) \] Veriﬁed OK.

Maple solution	\begin{align} y \left (x \right ) &= c_{1} \left (x -1\right ) \\ y \left (x \right ) &= c_{1} \left (x +1\right ) \\ \end{align}

Problem 2364


ODE	\[ \boxed {y^{\prime }-\sqrt {1-y}=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0] \end {align} With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = x -\frac {x^{2}}{4}+O\left (x^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = x -\frac {1}{4} x^{2} \]

Problem 2365


ODE	\[ \boxed {y^{\prime }-y x=-x^{2}} \] With initial conditions \begin {align} [y \left (0\right ) = 2] \end {align} With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = x^{2}+2-\frac {x^{3}}{3}+\frac {x^{4}}{4}+O\left (x^{5}\right ) \] Veriﬁed OK. \[ y = x^{2}+2-\frac {x^{3}}{3}+\frac {x^{4}}{4}+O\left (x^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 2+x^{2}-\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\operatorname {O}\left (x^{5}\right ) \]

Problem 2366


ODE	\[ \boxed {y^{\prime }-x^{2} y^{2}=0} \] With initial conditions \begin {align} [y \left (1\right ) = 0] \end {align} With the expansion point for the power series method at \(x = 1\).

program solution	\[ y = O\left (\left (x -1\right )^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 0 \]

Problem 2367


ODE	\[ \boxed {y^{\prime }-\frac {y}{x}=3 x} \] With initial conditions \begin {align} [y \left (1\right ) = 3] \end {align} With the expansion point for the power series method at \(x = 1\).

program solution	\[ y = 3 \left (x -1\right )^{2}+6 x -3+O\left (\left (x -1\right )^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 3 x^{2} \]

Problem 2368


ODE	\[ \boxed {y^{\prime }-\ln \left (y x \right )=0} \] With initial conditions \begin {align} [y \left (1\right ) = 1] \end {align} With the expansion point for the power series method at \(x = 1\).

program solution	\[ y = 1+\frac {\left (x -1\right )^{2}}{2}+\frac {\left (x -1\right )^{4}}{12}+O\left (\left (x -1\right )^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 1+\frac {1}{2} \left (x -1\right )^{2}+\frac {1}{12} \left (x -1\right )^{4}+\operatorname {O}\left (\left (x -1\right )^{5}\right ) \]

Problem 2369


ODE	\[ \boxed {y^{\prime }-y^{2}=1} \] With initial conditions \begin {align} [y \left (1\right ) = -1] \end {align} With the expansion point for the power series method at \(x = 1\).

program solution	\[ y = -2 \left (x -1\right )^{2}+2 x -3+\frac {8 \left (x -1\right )^{3}}{3}-\frac {10 \left (x -1\right )^{4}}{3}+O\left (\left (x -1\right )^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = -1+2 \left (x -1\right )-2 \left (x -1\right )^{2}+\frac {8}{3} \left (x -1\right )^{3}-\frac {10}{3} \left (x -1\right )^{4}+\operatorname {O}\left (\left (x -1\right )^{5}\right ) \]

Problem 2370


ODE	\[ \boxed {y^{\prime }-y^{2}=x^{2}} \] With initial conditions \begin {align} [y \left (2\right ) = 0] \end {align} With the expansion point for the power series method at \(x = 2\).

program solution	\[ y = 2 \left (-2+x \right )^{2}-8+4 x +\frac {17 \left (-2+x \right )^{3}}{3}+4 \left (-2+x \right )^{4}+\frac {148 \left (-2+x \right )^{5}}{15}+O\left (\left (-2+x \right )^{6}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 4 \left (-2+x \right )+2 \left (-2+x \right )^{2}+\frac {17}{3} \left (-2+x \right )^{3}+4 \left (-2+x \right )^{4}+\frac {148}{15} \left (-2+x \right )^{5}+\operatorname {O}\left (\left (-2+x \right )^{6}\right ) \]

Problem 2371


ODE	\[ \boxed {y^{\prime }-\sqrt {1+y x}=0} \] With initial conditions \begin {align} [y \left (0\right ) = 1] \end {align} With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = x +1+\frac {x^{2}}{4}+\frac {x^{3}}{8}+O\left (x^{4}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 1+x +\frac {1}{4} x^{2}+\frac {1}{8} x^{3}+\operatorname {O}\left (x^{4}\right ) \]

Problem 2372


ODE	\[ \boxed {y^{\prime }-\sin \left (y\right )=\cos \left (x \right )} \] With initial conditions \begin {align} \left [y \left (\frac {\pi }{2}\right ) = \frac {\pi }{2}\right ] \end {align} With the expansion point for the power series method at \(x = \frac {\pi }{2}\).

program solution	\[ y = x -\frac {\left (x -\frac {\pi }{2}\right )^{2}}{2}-\frac {\left (x -\frac {\pi }{2}\right )^{3}}{6}+O\left (\left (x -\frac {\pi }{2}\right )^{4}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {\pi }{2}+\left (-\frac {\pi }{2}+x \right )-\frac {1}{2} \left (-\frac {\pi }{2}+x \right )^{2}-\frac {1}{6} \left (-\frac {\pi }{2}+x \right )^{3}+\operatorname {O}\left (\left (-\frac {\pi }{2}+x \right )^{4}\right ) \]

Problem 2373


ODE	\[ \boxed {y^{\prime \prime }-y=\sin \left (x \right )} \] With initial conditions \begin {align} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align} With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = 1+2 x +\frac {x^{2}}{2}+\frac {x^{3}}{2}+\frac {x^{4}}{24}+\frac {x^{5}}{60}+\frac {x^{6}}{720}+\frac {x^{7}}{1680}+O\left (x^{7}\right ) \] Veriﬁed OK. \[ y = 1+\frac {x^{2}}{2}+\frac {x^{4}}{24}+\frac {x^{6}}{720}+2 x +\frac {x^{3}}{2}+\frac {x^{5}}{60}+O\left (x^{7}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 1+2 x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {1}{24} x^{4}+\frac {1}{60} x^{5}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{7}\right ) \]

Problem 2374


ODE	\[ \boxed {y^{\prime \prime }-2 y={\mathrm e}^{2 x}} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align} With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = \frac {x^{2}}{2}+\frac {x^{3}}{3}+\frac {x^{4}}{4}+\frac {x^{5}}{10}+\frac {7 x^{6}}{180}+\frac {x^{7}}{90}+O\left (x^{7}\right ) \] Veriﬁed OK. \[ y = \frac {x^{2}}{2}+\frac {x^{3}}{3}+\frac {x^{4}}{4}+\frac {x^{5}}{10}+\frac {7 x^{6}}{180}+O\left (x^{7}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {1}{2} x^{2}+\frac {1}{3} x^{3}+\frac {1}{4} x^{4}+\frac {1}{10} x^{5}+\frac {7}{180} x^{6}+\operatorname {O}\left (x^{7}\right ) \]

Problem 2375


ODE	\[ \boxed {y^{\prime \prime }+2 y y^{\prime }=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align} With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = x -\frac {x^{3}}{3}+\frac {2 x^{5}}{15}-\frac {17 x^{7}}{315}+O\left (x^{7}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = x -\frac {1}{3} x^{3}+\frac {2}{15} x^{5}+\operatorname {O}\left (x^{7}\right ) \]

Problem 2376


ODE	\[ \boxed {y^{\prime \prime }-\sin \left (y\right )=0} \] With initial conditions \begin {align} \left [y \left (0\right ) = \frac {\pi }{4}, y^{\prime }\left (0\right ) = 0\right ] \end {align} With the expansion point for the power series method at \(x = \frac {\pi }{4}\).

program solution	\[ y = \frac {\pi }{4}+\frac {\sqrt {2}\, \left (x -\frac {\pi }{4}\right )^{2}}{4}+\frac {\left (x -\frac {\pi }{4}\right )^{4}}{48}-\frac {\sqrt {2}\, \left (x -\frac {\pi }{4}\right )^{6}}{1440}+O\left (\left (x -\frac {\pi }{4}\right )^{7}\right ) \] Veriﬁed OK.

Maple solution	\[ \text {No solution found} \]

Problem 2377


ODE	\[ \boxed {y^{\prime \prime }+\frac {{y^{\prime }}^{2}}{2}-y=0} \] With initial conditions \begin {align} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align} With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = x -\frac {x^{2}}{4}+\frac {x^{3}}{4}-\frac {3 x^{4}}{32}+\frac {x^{5}}{20}-\frac {13 x^{6}}{480}+\frac {33 x^{7}}{2240}+O\left (x^{7}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = x -\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {3}{32} x^{4}+\frac {1}{20} x^{5}-\frac {13}{480} x^{6}+\operatorname {O}\left (x^{7}\right ) \]

Problem 2378


ODE	\[ \boxed {y^{\prime \prime }-\sin \left (y x \right )=0} \] With initial conditions \begin {align} \left [y \left (\frac {\pi }{2}\right ) = 1, y^{\prime }\left (\frac {\pi }{2}\right ) = 1\right ] \end {align} With the expansion point for the power series method at \(x = \frac {\pi }{2}\).

program solution	\[ y = x -\frac {\pi }{2}+1+\frac {\left (x -\frac {\pi }{2}\right )^{2}}{2}-\frac {\left (x -\frac {\pi }{2}\right )^{4} \pi ^{2}}{96}-\frac {\left (x -\frac {\pi }{2}\right )^{4} \pi }{24}-\frac {\left (x -\frac {\pi }{2}\right )^{4}}{24}-\frac {\left (x -\frac {\pi }{2}\right )^{5} \pi ^{2}}{160}-\frac {3 \left (x -\frac {\pi }{2}\right )^{5} \pi }{80}-\frac {\left (x -\frac {\pi }{2}\right )^{5}}{20}+O\left (\left (x -\frac {\pi }{2}\right )^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 1+\left (-\frac {\pi }{2}+x \right )+\frac {1}{2} \left (-\frac {\pi }{2}+x \right )^{2}-\frac {1}{96} \left (\pi +2\right )^{2} \left (-\frac {\pi }{2}+x \right )^{4}+\operatorname {O}\left (\left (-\frac {\pi }{2}+x \right )^{5}\right ) \]

Problem 2379


ODE	\[ \boxed {y^{\prime \prime }-\cos \left (y x \right )=0} \] With initial conditions \begin {align} \left [y \left (\frac {\pi }{2}\right ) = 1, y^{\prime }\left (\frac {\pi }{2}\right ) = 1\right ] \end {align} With the expansion point for the power series method at \(x = \frac {\pi }{2}\).

program solution	\[ y = x -\frac {\pi }{2}+1-\frac {\left (x -\frac {\pi }{2}\right )^{3} \pi }{12}-\frac {\left (x -\frac {\pi }{2}\right )^{3}}{6}-\frac {\left (x -\frac {\pi }{2}\right )^{4}}{12}+\frac {\left (x -\frac {\pi }{2}\right )^{5} \pi ^{3}}{960}+\frac {\left (x -\frac {\pi }{2}\right )^{5} \pi ^{2}}{120}+\frac {\left (x -\frac {\pi }{2}\right )^{5} \pi }{60}+\frac {\left (x -\frac {\pi }{2}\right )^{5}}{120}+O\left (\left (x -\frac {\pi }{2}\right )^{5}\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = 1+\left (-\frac {\pi }{2}+x \right )+\left (-\frac {\pi }{12}-\frac {1}{6}\right ) \left (-\frac {\pi }{2}+x \right )^{3}-\frac {1}{12} \left (-\frac {\pi }{2}+x \right )^{4}+\operatorname {O}\left (\left (-\frac {\pi }{2}+x \right )^{5}\right ) \]

Problem 2380


ODE	\[ \boxed {2 x y^{\prime \prime }+5 y^{\prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} \left (1-\frac {x^{2}}{14}+\frac {x^{4}}{616}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{40}+O\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}}+c_{2} \left (1-\frac {1}{14} x^{2}+\frac {1}{616} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2381


ODE	\[ \boxed {3 x \left (2+3 x \right ) y^{\prime \prime }-4 y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{\frac {5}{3}} \left (1-\frac {7 x}{8}+\frac {7 x^{2}}{8}-\frac {23 x^{3}}{24}+\frac {1817 x^{4}}{1632}-\frac {219857 x^{5}}{163200}+O\left (x^{6}\right )\right )+c_{2} \left (1+x -x^{2}+\frac {11 x^{3}}{12}-\frac {319 x^{4}}{336}+\frac {319 x^{5}}{300}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {5}{3}} \left (1-\frac {7}{8} x +\frac {7}{8} x^{2}-\frac {23}{24} x^{3}+\frac {1817}{1632} x^{4}-\frac {219857}{163200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+x -x^{2}+\frac {11}{12} x^{3}-\frac {319}{336} x^{4}+\frac {319}{300} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2382


ODE	\[ \boxed {x^{2} \left (4+x \right ) y^{\prime \prime }+7 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{\frac {1}{4}} \left (1+\frac {x}{48}-\frac {5 x^{2}}{19968}+\frac {25 x^{3}}{1810432}-\frac {75 x^{4}}{62390272}+\frac {39 x^{5}}{293601280}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+2 x +O\left (x^{6}\right )\right )}{x} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{4}} \left (1+\frac {1}{48} x -\frac {5}{19968} x^{2}+\frac {25}{1810432} x^{3}-\frac {75}{62390272} x^{4}+\frac {39}{293601280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+2 x +\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2383


ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+\left (-x^{2}+x \right ) y^{\prime }-y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x \left (1+\frac {x}{5}+\frac {x^{2}}{35}+\frac {x^{3}}{315}+\frac {x^{4}}{3465}+\frac {x^{5}}{45045}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {x}{2}+\frac {x^{2}}{8}+\frac {x^{3}}{48}+\frac {x^{4}}{384}+\frac {x^{5}}{3840}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x +\frac {1}{35} x^{2}+\frac {1}{315} x^{3}+\frac {1}{3465} x^{4}+\frac {1}{45045} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2384


ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+5 y^{\prime } x +y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = \frac {c_{1} \left (1-\frac {x}{3}+\frac {x^{2}}{30}-\frac {x^{3}}{630}+\frac {x^{4}}{22680}-\frac {x^{5}}{1247400}+O\left (x^{6}\right )\right )}{\sqrt {x}}+\frac {c_{2} \left (1-x +\frac {x^{2}}{6}-\frac {x^{3}}{90}+\frac {x^{4}}{2520}-\frac {x^{5}}{113400}+O\left (x^{6}\right )\right )}{x} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

Problem 2385


ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+\left (2+3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{\frac {2}{3}} \left (1-\frac {x}{4}+\frac {x^{2}}{56}-\frac {x^{3}}{1680}+\frac {x^{4}}{87360}-\frac {x^{5}}{6988800}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{3}} \left (1-\frac {x}{2}+\frac {x^{2}}{20}-\frac {x^{3}}{480}+\frac {x^{4}}{21120}-\frac {x^{5}}{1478400}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{3}} \left (1-\frac {1}{2} x +\frac {1}{20} x^{2}-\frac {1}{480} x^{3}+\frac {1}{21120} x^{4}-\frac {1}{1478400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\frac {2}{3}} \left (1-\frac {1}{4} x +\frac {1}{56} x^{2}-\frac {1}{1680} x^{3}+\frac {1}{87360} x^{4}-\frac {1}{6988800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2386


ODE	\[ \boxed {\left (x^{3}+2 x^{2}\right ) y^{\prime \prime }-y^{\prime } x +\left (1-x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x \left (1+\frac {x}{3}-\frac {x^{2}}{30}+\frac {x^{3}}{126}-\frac {11 x^{4}}{4536}+\frac {19 x^{5}}{22680}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {5 x}{4}+\frac {5 x^{2}}{96}-\frac {11 x^{3}}{1152}+\frac {341 x^{4}}{129024}-\frac {20119 x^{5}}{23224320}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {5}{4} x +\frac {5}{96} x^{2}-\frac {11}{1152} x^{3}+\frac {341}{129024} x^{4}-\frac {20119}{23224320} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{3} x -\frac {1}{30} x^{2}+\frac {1}{126} x^{3}-\frac {11}{4536} x^{4}+\frac {19}{22680} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2387


ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-3 \left (x^{2}+x \right ) y^{\prime }+\left (2+3 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{2} \left (1+\frac {3 x}{5}+\frac {9 x^{2}}{35}+\frac {3 x^{3}}{35}+\frac {9 x^{4}}{385}+\frac {27 x^{5}}{5005}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {3 x}{2}+\frac {9 x^{2}}{8}+\frac {9 x^{3}}{16}+\frac {27 x^{4}}{128}+\frac {81 x^{5}}{1280}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {3}{2} x +\frac {9}{8} x^{2}+\frac {9}{16} x^{3}+\frac {27}{128} x^{4}+\frac {81}{1280} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1+\frac {3}{5} x +\frac {9}{35} x^{2}+\frac {3}{35} x^{3}+\frac {9}{385} x^{4}+\frac {27}{5005} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2388


ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+\left (-x^{2}+5 x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {x}{21}-\frac {61 x^{2}}{630}-\frac {607 x^{3}}{73710}+\frac {2297 x^{4}}{884520}+\frac {14713 x^{5}}{50417640}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+x -\frac {x^{2}}{2}-\frac {x^{3}}{6}+\frac {x^{4}}{48}+\frac {19 x^{5}}{2640}+O\left (x^{6}\right )\right )}{x} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {4}{3}} \left (1+\frac {1}{21} x -\frac {61}{630} x^{2}-\frac {607}{73710} x^{3}+\frac {2297}{884520} x^{4}+\frac {14713}{50417640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+x -\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{48} x^{4}+\frac {19}{2640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

Problem 2389


ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+x \left (x^{2}-4\right ) y^{\prime }+3 y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{\frac {3}{2}} \left (1-\frac {x^{2}}{16}+\frac {7 x^{4}}{2560}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {x^{2}}{16}+\frac {5 x^{4}}{1536}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \sqrt {x}\, \left (x \left (1-\frac {1}{16} x^{2}+\frac {7}{2560} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{1} +\left (1-\frac {1}{16} x^{2}+\frac {5}{1536} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

Problem 2390


ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-3 \left (x^{2}+x \right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{\frac {7}{8}+\frac {\sqrt {17}}{8}} \left (1+\frac {\left (21+3 \sqrt {17}\right ) x}{32+8 \sqrt {17}}+\frac {9 \left (7+\sqrt {17}\right ) \left (15+\sqrt {17}\right ) x^{2}}{128 \left (4+\sqrt {17}\right ) \left (8+\sqrt {17}\right )}+\frac {9 \left (7+\sqrt {17}\right ) \left (15+\sqrt {17}\right ) \left (23+\sqrt {17}\right ) x^{3}}{1024 \left (49+12 \sqrt {17}\right ) \left (12+\sqrt {17}\right )}+\frac {27 \left (7+\sqrt {17}\right ) \left (15+\sqrt {17}\right ) \left (23+\sqrt {17}\right ) \left (31+\sqrt {17}\right ) x^{4}}{32768 \left (792+193 \sqrt {17}\right ) \left (16+\sqrt {17}\right )}+\frac {81 \left (7+\sqrt {17}\right ) \left (15+\sqrt {17}\right ) \left (23+\sqrt {17}\right ) \left (31+\sqrt {17}\right ) \left (39+\sqrt {17}\right ) x^{5}}{1310720 \left (15953+3880 \sqrt {17}\right ) \left (20+\sqrt {17}\right )}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {7}{8}-\frac {\sqrt {17}}{8}} \left (1+\frac {\left (-21+3 \sqrt {17}\right ) x}{-32+8 \sqrt {17}}+\frac {9 \left (-7+\sqrt {17}\right ) \left (-15+\sqrt {17}\right ) x^{2}}{128 \left (-4+\sqrt {17}\right ) \left (-8+\sqrt {17}\right )}-\frac {9 \left (-7+\sqrt {17}\right ) \left (-15+\sqrt {17}\right ) \left (-23+\sqrt {17}\right ) x^{3}}{1024 \left (-49+12 \sqrt {17}\right ) \left (-12+\sqrt {17}\right )}+\frac {27 \left (-7+\sqrt {17}\right ) \left (-15+\sqrt {17}\right ) \left (-23+\sqrt {17}\right ) \left (-31+\sqrt {17}\right ) x^{4}}{32768 \left (-792+193 \sqrt {17}\right ) \left (-16+\sqrt {17}\right )}-\frac {81 \left (-7+\sqrt {17}\right ) \left (-15+\sqrt {17}\right ) \left (-23+\sqrt {17}\right ) \left (-31+\sqrt {17}\right ) \left (-39+\sqrt {17}\right ) x^{5}}{1310720 \left (-15953+3880 \sqrt {17}\right ) \left (-20+\sqrt {17}\right )}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = x^{\frac {7}{8}} \left (c_{2} x^{\frac {\sqrt {17}}{8}} \left (1+\frac {21+3 \sqrt {17}}{8 \sqrt {17}+32} x +\frac {9}{128} \frac {\left (15+\sqrt {17}\right ) \left (7+\sqrt {17}\right )}{\left (4+\sqrt {17}\right ) \left (8+\sqrt {17}\right )} x^{2}+\frac {9}{1024} \frac {\left (23+\sqrt {17}\right ) \left (15+\sqrt {17}\right ) \left (7+\sqrt {17}\right )}{\left (4+\sqrt {17}\right ) \left (8+\sqrt {17}\right ) \left (12+\sqrt {17}\right )} x^{3}+\frac {27}{32768} \frac {\left (31+\sqrt {17}\right ) \left (23+\sqrt {17}\right ) \left (15+\sqrt {17}\right ) \left (7+\sqrt {17}\right )}{\left (4+\sqrt {17}\right ) \left (8+\sqrt {17}\right ) \left (12+\sqrt {17}\right ) \left (16+\sqrt {17}\right )} x^{4}+\frac {81}{1310720} \frac {\left (39+\sqrt {17}\right ) \left (31+\sqrt {17}\right ) \left (23+\sqrt {17}\right ) \left (15+\sqrt {17}\right ) \left (7+\sqrt {17}\right )}{\left (4+\sqrt {17}\right ) \left (8+\sqrt {17}\right ) \left (12+\sqrt {17}\right ) \left (16+\sqrt {17}\right ) \left (20+\sqrt {17}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} x^{-\frac {\sqrt {17}}{8}} \left (1+\frac {-21+3 \sqrt {17}}{8 \sqrt {17}-32} x +\frac {9}{128} \frac {\left (-15+\sqrt {17}\right ) \left (-7+\sqrt {17}\right )}{\left (-4+\sqrt {17}\right ) \left (-8+\sqrt {17}\right )} x^{2}+\frac {9}{1024} \frac {\left (-23+\sqrt {17}\right ) \left (-15+\sqrt {17}\right ) \left (-7+\sqrt {17}\right )}{\left (-4+\sqrt {17}\right ) \left (-8+\sqrt {17}\right ) \left (-12+\sqrt {17}\right )} x^{3}+\frac {27}{32768} \frac {\left (-31+\sqrt {17}\right ) \left (-23+\sqrt {17}\right ) \left (-15+\sqrt {17}\right ) \left (-7+\sqrt {17}\right )}{\left (-4+\sqrt {17}\right ) \left (-8+\sqrt {17}\right ) \left (-12+\sqrt {17}\right ) \left (-16+\sqrt {17}\right )} x^{4}+\frac {81}{1310720} \frac {\left (-39+\sqrt {17}\right ) \left (-31+\sqrt {17}\right ) \left (-23+\sqrt {17}\right ) \left (-15+\sqrt {17}\right ) \left (-7+\sqrt {17}\right )}{\left (-4+\sqrt {17}\right ) \left (-8+\sqrt {17}\right ) \left (-12+\sqrt {17}\right ) \left (-16+\sqrt {17}\right ) \left (-20+\sqrt {17}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

Problem 2391


ODE	\[ \boxed {9 x^{2} y^{\prime \prime }+9 \left (-x^{2}+x \right ) y^{\prime }+y \left (x -1\right )=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{\frac {1}{3}} \left (1+\frac {2 x}{15}+\frac {11 x^{2}}{360}+\frac {x^{3}}{162}+\frac {29 x^{4}}{27216}+\frac {551 x^{5}}{3470040}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {4 x}{3}-\frac {5 x^{2}}{18}-\frac {5 x^{3}}{81}-\frac {23 x^{4}}{1944}-\frac {92 x^{5}}{47385}+O\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {2}{3}} \left (1+\frac {2}{15} x +\frac {11}{360} x^{2}+\frac {1}{162} x^{3}+\frac {29}{27216} x^{4}+\frac {551}{3470040} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {4}{3} x -\frac {5}{18} x^{2}-\frac {5}{81} x^{3}-\frac {23}{1944} x^{4}-\frac {92}{47385} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{3}}} \]

Problem 2392


ODE	\[ \boxed {4 x^{2} \left (1-x \right ) y^{\prime \prime }+3 x \left (1+2 x \right ) y^{\prime }-3 y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x \left (1-\frac {6 x}{11}+\frac {4 x^{2}}{55}+\frac {8 x^{3}}{1045}+\frac {48 x^{4}}{24035}+\frac {32 x^{5}}{43263}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {13 x}{4}+\frac {117 x^{2}}{32}-\frac {195 x^{3}}{128}+\frac {195 x^{4}}{2048}+\frac {117 x^{5}}{8192}+O\left (x^{6}\right )\right )}{x^{\frac {3}{4}}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {13}{4} x +\frac {117}{32} x^{2}-\frac {195}{128} x^{3}+\frac {195}{2048} x^{4}+\frac {117}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {3}{4}}}+c_{2} x \left (1-\frac {6}{11} x +\frac {4}{55} x^{2}+\frac {8}{1045} x^{3}+\frac {48}{24035} x^{4}+\frac {32}{43263} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2393


ODE	\[ \boxed {2 x^{2} \left (1-3 x \right ) y^{\prime \prime }+5 y^{\prime } x -2 y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {3 x}{14}-\frac {3 x^{2}}{56}-\frac {45 x^{3}}{1232}-\frac {675 x^{4}}{18304}-\frac {1701 x^{5}}{36608}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-12 x +72 x^{2}+O\left (x^{6}\right )\right )}{x^{2}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {3}{14} x -\frac {3}{56} x^{2}-\frac {45}{1232} x^{3}-\frac {675}{18304} x^{4}-\frac {1701}{36608} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1-12 x +72 x^{2}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

Problem 2394


ODE	\[ \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }-5 y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x^{2} \left (1-\frac {8 x}{11}+\frac {32 x^{2}}{55}-\frac {512 x^{3}}{1045}+\frac {2048 x^{4}}{4807}-\frac {16384 x^{5}}{43263}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{4}+\frac {5 x^{2}}{32}-\frac {15 x^{3}}{128}+\frac {195 x^{4}}{2048}-\frac {663 x^{5}}{8192}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{4} x +\frac {5}{32} x^{2}-\frac {15}{128} x^{3}+\frac {195}{2048} x^{4}-\frac {663}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1-\frac {8}{11} x +\frac {32}{55} x^{2}-\frac {512}{1045} x^{3}+\frac {2048}{4807} x^{4}-\frac {16384}{43263} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2395


ODE	\[ \boxed {\left (4+x \right ) x^{2} y^{\prime \prime }+x \left (x -1\right ) y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x \left (1-\frac {x}{7}+\frac {2 x^{2}}{77}-\frac {2 x^{3}}{385}+\frac {8 x^{4}}{7315}-\frac {8 x^{5}}{33649}+O\left (x^{6}\right )\right )+c_{2} x^{\frac {1}{4}} \left (1-\frac {x}{16}+\frac {5 x^{2}}{512}-\frac {15 x^{3}}{8192}+\frac {195 x^{4}}{524288}-\frac {663 x^{5}}{8388608}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {1}{16} x +\frac {5}{512} x^{2}-\frac {15}{8192} x^{3}+\frac {195}{524288} x^{4}-\frac {663}{8388608} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1-\frac {1}{7} x +\frac {2}{77} x^{2}-\frac {2}{385} x^{3}+\frac {8}{7315} x^{4}-\frac {8}{33649} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2396


ODE	\[ \boxed {\left (8-x \right ) x^{2} y^{\prime \prime }+6 y^{\prime } x -y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{56}-\frac {3 x^{2}}{9856}-\frac {x^{3}}{78848}-\frac {5 x^{4}}{6848512}-\frac {63 x^{5}}{1260126208}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1+\frac {5 x}{32}-\frac {3 x^{2}}{2048}-\frac {7 x^{3}}{196608}-\frac {539 x^{4}}{327155712}-\frac {5929 x^{5}}{59324235776}+O\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{\frac {3}{4}} \left (1-\frac {1}{56} x -\frac {3}{9856} x^{2}-\frac {1}{78848} x^{3}-\frac {5}{6848512} x^{4}-\frac {63}{1260126208} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+\frac {5}{32} x -\frac {3}{2048} x^{2}-\frac {7}{196608} x^{3}-\frac {539}{327155712} x^{4}-\frac {5929}{59324235776} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {1}{4}}} \]

Problem 2397


ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+x \left (x^{2}+1\right ) y^{\prime }-y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x \left (1+\frac {x}{5}-\frac {2 x^{2}}{35}-\frac {16 x^{3}}{945}+\frac {73 x^{4}}{20790}+\frac {1481 x^{5}}{1351350}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-x -\frac {x^{2}}{4}+\frac {x^{3}}{36}+\frac {29 x^{4}}{1440}-\frac {71 x^{5}}{50400}+O\left (x^{6}\right )\right )}{\sqrt {x}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-x -\frac {1}{4} x^{2}+\frac {1}{36} x^{3}+\frac {29}{1440} x^{4}-\frac {71}{50400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_{2} x \left (1+\frac {1}{5} x -\frac {2}{35} x^{2}-\frac {16}{945} x^{3}+\frac {73}{20790} x^{4}+\frac {1481}{1351350} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2398


ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-y^{\prime } x +\left (x^{2}+1\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1-\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2399


ODE	\[ \boxed {3 x^{2} y^{\prime \prime }+2 y^{\prime } x +\left (x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	\[ y = c_{1} x \left (1-\frac {x^{2}}{22}+\frac {x^{4}}{1496}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {x^{2}}{2}+\frac {x^{4}}{56}+O\left (x^{6}\right )\right )}{x^{\frac {2}{3}}} \] Veriﬁed OK.

Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{56} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{\frac {2}{3}}}+c_{2} x \left (1-\frac {1}{22} x^{2}+\frac {1}{1496} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 2400


ODE	\[ \boxed {x^{3} \left (x^{2}+3\right ) y^{\prime \prime }+5 y^{\prime } x -y \left (x +1\right )=0} \] With the expansion point for the power series method at \(x = 0\).

program solution	N/A

Maple solution	\[ \text {No solution found} \]

2.17.24 Problems 2301 to 2400