2.17.117 Problems 11601 to 11700

Problem 11601

ODE	\[ \boxed {\frac {x}{y^{2}}+\left (\frac {x^{2}}{y^{3}}+y\right ) y^{\prime }=-x} \]
program solution	\[ \frac {x^{2} y^{2}}{2 y^{2}+2}+\frac {y^{2}}{2}-\ln \left (1+y^{2}\right )-\frac {1}{2 \left (1+y^{2}\right )} = c_{1} \] Verified OK.
Maple solution	\[ \frac {\left (-2 y \left (x \right )^{2}-2\right ) \ln \left (y \left (x \right )^{2}+1\right )+y \left (x \right )^{4}+\left (x^{2}+2 c_{1} +1\right ) y \left (x \right )^{2}+2 c_{1} -1}{2 y \left (x \right )^{2}+2} = 0 \]

Problem 11602

ODE	\[ \boxed {\frac {\left (2 s-1\right ) s^{\prime }}{t}+\frac {s-s^{2}}{t^{2}}=0} \]
program solution	\[ -\ln \left (t \right )+\ln \left (s \left (s-1\right )\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} s \left (t \right ) &= \frac {1}{2}-\frac {\sqrt {4 c_{1} t +1}}{2} \\ s \left (t \right ) &= \frac {1}{2}+\frac {\sqrt {4 c_{1} t +1}}{2} \\ \end{align*}

Problem 11603

ODE	\[ \boxed {\frac {2 y^{\frac {3}{2}}+1}{x^{\frac {1}{3}}}+\left (3 \sqrt {x}\, \sqrt {y}-1\right ) y^{\prime }=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 11604

ODE	\[ \boxed {2 x y+\left (x^{2}+4 y\right ) y^{\prime }=3} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ x^{2} y+2 y^{2}-3 x = 7 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x^{2}}{4}+\frac {\sqrt {x^{4}+24 x +56}}{4} \]

Problem 11605

ODE	\[ \boxed {3 x^{2} y^{2}-y^{3}+\left (2 y x^{3}-3 y^{2} x +1\right ) y^{\prime }=-2 x} \] With initial conditions \begin {align*} [y \left (-2\right ) = 1] \end {align*}
program solution	\[ y^{2} x^{3}-y^{3} x +x^{2}+y = -1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{\frac {2}{3}}}{2}+x \left (2 x^{2} {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{\frac {1}{3}}+\left (x^{5}+3\right ) 2^{\frac {1}{3}} \left (i \sqrt {3}-1\right )\right )}{6 {\left (\left (2 x^{7}+3 \sqrt {3}\, \sqrt {\frac {4 x^{10}+4 x^{8}+44 x^{5}+72 x^{3}+27 x -4}{x}}+36 x^{2}+27\right ) x^{2}\right )}^{\frac {1}{3}} x} \]

Problem 11606

ODE	\[ \boxed {2 \sin \left (x \right ) y \cos \left (x \right )+\sin \left (x \right ) y^{2}+\left (\sin \left (x \right )^{2}-2 y \cos \left (x \right )\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ -y \left (y \cos \left (x \right )-\sin \left (x \right )^{2}\right ) = -9 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sec \left (x \right ) \left (\sin \left (x \right )^{2}+\sqrt {\sin \left (x \right )^{4}+36 \cos \left (x \right )}\right )}{2} \]

Problem 11607

ODE	\[ \boxed {y \,{\mathrm e}^{x}+y^{2}+\left ({\mathrm e}^{x}+2 x y\right ) y^{\prime }=-2 \,{\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 6] \end {align*}
program solution	\[ {\mathrm e}^{x} \left (y+2\right )+y^{2} x = 8 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-{\mathrm e}^{x}+\sqrt {{\mathrm e}^{2 x}-8 \,{\mathrm e}^{x} x +32 x}}{2 x} \]

Problem 11608

ODE	\[ \boxed {\frac {3-y}{x^{2}}+\frac {\left (y^{2}-2 x \right ) y^{\prime }}{y^{2} x}=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 2] \end {align*}
program solution	\[ \frac {y^{2}+2 x -3 y}{y x} = 2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = x +\frac {3}{2}+\frac {\sqrt {4 x^{2}+4 x +9}}{2} \]

Problem 11609

ODE	\[ \boxed {\frac {1+8 x y^{\frac {2}{3}}}{x^{\frac {2}{3}} y^{\frac {1}{3}}}+\frac {\left (2 x^{\frac {4}{3}} y^{\frac {2}{3}}-x^{\frac {1}{3}}\right ) y^{\prime }}{y^{\frac {4}{3}}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 8] \end {align*}
program solution	\[ \frac {6 x^{\frac {1}{3}} \left (x y^{\frac {2}{3}}+\frac {1}{2}\right )}{y^{\frac {1}{3}}} = {\frac {27}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (64 \textit {\_Z}^{\frac {7}{3}} x^{4}+96 \textit {\_Z}^{\frac {5}{3}} x^{3}-729 \textit {\_Z}^{\frac {4}{3}}+48 x^{2} \textit {\_Z} +8 x \,\textit {\_Z}^{\frac {1}{3}}\right ) \]

Problem 11610

ODE	\[ \boxed {3 y^{2}+2 y y^{\prime } x=-4 x} \]
program solution	\[ x^{3} \left (y^{2}+x \right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\sqrt {x \left (-x^{4}+c_{1} \right )}}{x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {x \left (-x^{4}+c_{1} \right )}}{x^{2}} \\ \end{align*}

Problem 11611

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

Problem 11612

ODE	\[ \boxed {y+x \left (x^{2}+y^{2}\right )^{2}+\left (y \left (x^{2}+y^{2}\right )^{2}-x \right ) y^{\prime }=0} \]
program solution	\[ \frac {x^{4}}{4}+\frac {x^{2} y^{2}}{2}+\arctan \left (\frac {x}{y}\right )+\frac {y^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \cot \left (\operatorname {RootOf}\left (4 c_{1} \sin \left (\textit {\_Z} \right )^{4}-4 \textit {\_Z} \sin \left (\textit {\_Z} \right )^{4}-x^{4}\right )\right ) x \]

Problem 11613

ODE	\[ \boxed {4 x y+\left (x^{2}+1\right ) y^{\prime }=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-4 c_{1}}}{\left (x^{2}+1\right )^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{\left (x^{2}+1\right )^{2}} \]

Problem 11614

ODE	\[ \boxed {x y+y+\left (x^{2}+2 x \right ) y^{\prime }=-2 x -2} \]
program solution	\[ y = {\mathrm e}^{-\frac {\ln \left (x \left (x +2\right )\right )}{2}-c_{1}}-2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+\frac {c_{1}}{\sqrt {x \left (x +2\right )}} \]

Problem 11615

ODE	\[ \boxed {2 r \left (s^{2}+1\right )+\left (r^{4}+1\right ) s^{\prime }=0} \]
program solution	\[ s = \frac {-r^{2}+c_{3}}{c_{3} r^{2}+1} \] Verified OK.
Maple solution	\[ s \left (r \right ) = -\tan \left (\arctan \left (r^{2}\right )+2 c_{1} \right ) \]

Problem 11616

ODE	\[ \boxed {\csc \left (y\right )+y^{\prime } \sec \left (x \right )=0} \]
program solution	\[ -\sin \left (x \right )+\cos \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arccos \left (\sin \left (x \right )+c_{1} \right ) \]

Problem 11617

ODE	\[ \boxed {\tan \left (\theta \right )+2 r \theta ^{\prime }=0} \]
program solution	\[ -\ln \left (r \right )-2 \ln \left (\sin \left (\theta \right )\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} \theta \left (r \right ) &= \arcsin \left (\frac {1}{\sqrt {c_{1} r}}\right ) \\ \theta \left (r \right ) &= -\arcsin \left (\frac {1}{\sqrt {c_{1} r}}\right ) \\ \end{align*}

Problem 11618

ODE	\[ \boxed {\left ({\mathrm e}^{v}+1\right ) \cos \left (u \right )+{\mathrm e}^{v} \left (1+\sin \left (u \right )\right ) v^{\prime }=0} \]
program solution	\[ v = \ln \left (-\frac {-1+\sin \left (u \right ) {\mathrm e}^{c_{1}}+{\mathrm e}^{c_{1}}}{1+\sin \left (u \right )}\right )-c_{1} \] Verified OK.
Maple solution	\[ v \left (u \right ) = -\ln \left (\frac {-1-\sin \left (u \right )}{-1+\left (1+\sin \left (u \right )\right ) {\mathrm e}^{c_{1}}}\right )-c_{1} \]

Problem 11619

ODE	\[ \boxed {\left (x +4\right ) \left (1+y^{2}\right )+y \left (x^{2}+3 x +2\right ) y^{\prime }=0} \]
program solution	\[ -3 \ln \left (x +1\right )+2 \ln \left (x +2\right )-\frac {\ln \left (1+y^{2}\right )}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\sqrt {-x^{6}-6 x^{5}+x^{4} c_{1} +\left (8 c_{1} +100\right ) x^{3}+\left (24 c_{1} +345\right ) x^{2}+\left (32 c_{1} +474\right ) x +16 c_{1} +239}}{\left (1+x \right )^{3}} \\ y \left (x \right ) &= -\frac {\sqrt {-x^{6}-6 x^{5}+x^{4} c_{1} +\left (8 c_{1} +100\right ) x^{3}+\left (24 c_{1} +345\right ) x^{2}+\left (32 c_{1} +474\right ) x +16 c_{1} +239}}{\left (1+x \right )^{3}} \\ \end{align*}

Problem 11620

ODE	\[ \boxed {-y^{\prime } x +y=-x} \]
program solution	\[ y = x \left (\ln \left (x \right )-c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\ln \left (x \right )+c_{1} \right ) x \]

Problem 11621

ODE	\[ \boxed {2 x y+3 y^{2}-\left (2 x y+x^{2}\right ) y^{\prime }=0} \]
program solution	\[ -\frac {y \left (y+x \right )}{x^{3}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (1+\sqrt {4 c_{1} x +1}\right ) x}{2} \\ y \left (x \right ) &= \frac {\left (-1+\sqrt {4 c_{1} x +1}\right ) x}{2} \\ \end{align*}

Problem 11622

ODE	\[ \boxed {v^{3}+\left (u^{3}-u v^{2}\right ) v^{\prime }=0} \]
program solution	\[ v = {\mathrm e}^{-\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{2 c_{1}}}{u^{2}}\right )}{2}+c_{1}} \] Verified OK.
Maple solution	\[ v \left (u \right ) = \frac {{\mathrm e}^{-c_{1}}}{\sqrt {-\frac {{\mathrm e}^{-2 c_{1}}}{u^{2} \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-2 c_{1}}}{u^{2}}\right )}}} \]

Problem 11623

ODE	\[ \boxed {x \tan \left (\frac {y}{x}\right )+y-y^{\prime } x=0} \]
program solution	\[ y = -\arcsin \left (c_{1} x \right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \arcsin \left (c_{1} x \right ) x \]

Problem 11624

ODE	\[ \boxed {\left (2 s^{2}+2 s t +t^{2}\right ) s^{\prime }+s^{2}+2 s t=t^{2}} \]
program solution	\[ s^{2} t +s t^{2}-\frac {t^{3}}{3}+\frac {2 s^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} s \left (t \right ) &= \frac {\left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}-\frac {t^{2} c_{1}^{2}}{\left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}}-c_{1} t}{2 c_{1}} \\ s \left (t \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {2}{3}}+c_{1} t \left (2 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) c_{1} t \right )}{4 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}} c_{1}} \\ s \left (t \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {2}{3}}+\left (-2 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}}+c_{1} t \left (1+i \sqrt {3}\right )\right ) c_{1} t}{4 \left (4 t^{3} c_{1}^{3}+2+\sqrt {17 c_{1}^{6} t^{6}+16 t^{3} c_{1}^{3}+4}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 11625

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

Problem 11626

ODE	\[ \boxed {\sqrt {y+x}+\sqrt {x -y}+\left (\sqrt {x -y}-\sqrt {y+x}\right ) y^{\prime }=0} \]
program solution	\[ \ln \left (x \right ) = \frac {i \sqrt {\frac {y}{x}+1}\, \sqrt {\frac {y}{x}-1}\, \arctan \left (\frac {1}{\sqrt {\frac {y^{2}}{x^{2}}-1}}\right )}{\sqrt {\frac {y^{2}}{x^{2}}-1}}-\ln \left (\frac {y}{x}\right )+c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right )+\ln \left (\frac {y \left (x \right )}{x}\right )-\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {-x^{2}+y \left (x \right )^{2}}{x^{2}}}}\right )-c_{1} = 0 \]

Problem 11627

ODE	\[ \boxed {y+y \left (x +4\right ) y^{\prime }=-2} \] With initial conditions \begin {align*} [y \left (-3\right ) = -1] \end {align*}
program solution	\[ y = -2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {1}{2}} \sqrt {x +4}}{2}\right )-2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \operatorname {LambertW}\left (-\frac {\sqrt {x +4}\, {\mathrm e}^{-\frac {1}{2}}}{2}\right )-2 \]

Problem 11628

ODE	\[ \boxed {8 \cos \left (y\right )^{2}+\csc \left (x \right )^{2} y^{\prime }=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{12}\right ) = \frac {\pi }{4}\right ] \end {align*}
program solution	\[ \frac {\sin \left (2 x \right )}{4}-\frac {x}{2}-\frac {\tan \left (y\right )}{8} = -\frac {\pi }{24} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\arctan \left (-\frac {\pi }{3}+4 x -2 \sin \left (2 x \right )\right ) \]

Problem 11629

ODE	\[ \boxed {\left (3 x +8\right ) \left (y^{2}+4\right )-4 y \left (x^{2}+5 x +6\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ -\ln \left (x +3\right )-2 \ln \left (x +2\right )+2 \ln \left (y^{2}+4\right ) = 4 \ln \left (2\right )-2 \ln \left (3\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \sqrt {-9+\left (3 x +6\right ) \sqrt {x +3}}}{3} \]

Problem 11630

ODE	\[ \boxed {3 y^{2}-2 y y^{\prime } x=-x^{2}} \] With initial conditions \begin {align*} [y \left (2\right ) = 6] \end {align*}
program solution	\[ \frac {-x^{2}-y^{2}}{x^{3}} = -5 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {5 x -1}\, x \]

Problem 11631

ODE	\[ \boxed {-5 y+\left (4 x -y\right ) y^{\prime }=-2 x} \] With initial conditions \begin {align*} [y \left (1\right ) = 4] \end {align*}
program solution	\[ 2 \ln \left (2 x +y\right )-\ln \left (y-x \right ) = 2 \ln \left (2\right )+\ln \left (3\right ) \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 6-2 x -6 \sqrt {1-x} \\ y \left (x \right ) &= 6-2 x +6 \sqrt {1-x} \\ \end{align*}

Problem 11632

ODE	\[ \boxed {9 x y+5 y^{2}-\left (6 x^{2}+4 x y\right ) y^{\prime }=-3 x^{2}} \] With initial conditions \begin {align*} [y \left (2\right ) = -6] \end {align*}
program solution	\[ -\frac {6 \left (x^{2}+x y+\frac {y^{2}}{3}\right )}{x^{\frac {5}{2}}} = -3 \sqrt {2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (3+\sqrt {-3+6 \sqrt {2}\, \sqrt {x}}\right ) x}{2} \]

Problem 11633

ODE	\[ \boxed {2 y+\left (2 x -y\right ) y^{\prime }=-x} \]
program solution	\[ \frac {x \left (4 y+x \right )}{2}-\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2 c_{1} x -\sqrt {5 c_{1}^{2} x^{2}+1}}{c_{1}} \\ y \left (x \right ) &= \frac {2 c_{1} x +\sqrt {5 c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*}

Problem 11634

ODE	\[ \boxed {-y-\left (y+x \right ) y^{\prime }=-3 x} \]
program solution	\[ \frac {x \left (3 x -2 y\right )}{2}-\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-c_{1} x -\sqrt {4 c_{1}^{2} x^{2}+1}}{c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x +\sqrt {4 c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*}

Problem 11635

ODE	\[ \boxed {2 y^{2}+\left (4 x y-y^{2}\right ) y^{\prime }=-x^{2}} \]
program solution	\[ \frac {x^{3}}{3}+2 y^{2} x -\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\frac {\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {8 x^{2} c_{1}^{2}}{\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x}{c_{1}} \\ y \left (x \right ) &= \frac {-\frac {\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {4 x^{2} c_{1}^{2}}{\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x -\frac {i \sqrt {3}\, \left (-16 c_{1}^{2} x^{2}+\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {2}{3}}\right )}{4 \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}}}{c_{1}} \\ y \left (x \right ) &= -\frac {16 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {2}{3}}+16 c_{1}^{2} x^{2}-8 c_{1} x \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}}+\left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {2}{3}}}{4 \left (4+68 c_{1}^{3} x^{3}+4 \sqrt {33 c_{1}^{6} x^{6}+34 c_{1}^{3} x^{3}+1}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 11636

ODE	\[ \boxed {2 x y+y^{2}+\left (2 x y+x^{2}\right ) y^{\prime }=-2 x^{2}} \]
program solution	\[ \frac {2 x^{3}}{3}+x^{2} y+y^{2} x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-3 c_{1}^{2} x^{2}+\sqrt {3}\, \sqrt {-5 \left (c_{1}^{3} x^{3}-\frac {4}{5}\right ) c_{1} x}}{6 c_{1}^{2} x} \\ y \left (x \right ) &= \frac {-3 c_{1}^{2} x^{2}-\sqrt {3}\, \sqrt {-5 \left (c_{1}^{3} x^{3}-\frac {4}{5}\right ) c_{1} x}}{6 c_{1}^{2} x} \\ \end{align*}

Problem 11637

ODE	\[ \boxed {y^{\prime }+\frac {3 y}{x}=6 x^{2}} \]
program solution	\[ y = \frac {x^{6}+c_{1}}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{6}+c_{1}}{x^{3}} \]

Problem 11638

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

Problem 11639

ODE	\[ \boxed {y^{\prime }+3 y=3 x^{2} {\mathrm e}^{-3 x}} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (x^{3}+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{3}+c_{1} \right ) {\mathrm e}^{-3 x} \]

Problem 11640

ODE	\[ \boxed {y^{\prime }+4 x y=8 x} \]
program solution	\[ y = {\mathrm e}^{-2 x^{2}-4 c_{1}}+2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+{\mathrm e}^{-2 x^{2}} c_{1} \]

Problem 11641

ODE	\[ \boxed {x^{\prime }+\frac {x}{t^{2}}=\frac {1}{t^{2}}} \]
program solution	\[ x = {\mathrm e}^{-\frac {c_{1} t -1}{t}}+1 \] Verified OK.
Maple solution	\[ x \left (t \right ) = 1+{\mathrm e}^{\frac {1}{t}} c_{1} \]

Problem 11642

ODE	\[ \boxed {\left (u^{2}+1\right ) v^{\prime }+4 v u=3 u} \]
program solution	\[ v = \frac {3 u^{4}+6 u^{2}+{\mathrm e}^{-4 c_{1}}+3}{4 u^{4}+8 u^{2}+4} \] Verified OK.
Maple solution	\[ v \left (u \right ) = \frac {3}{4}+\frac {c_{1}}{\left (u^{2}+1\right )^{2}} \]

Problem 11643

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

Problem 11644

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

Problem 11645

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

Problem 11646

ODE	\[ \boxed {y+\left (y^{2} x +x -y\right ) y^{\prime }=0} \]
program solution	\[ y \,{\mathrm e}^{\frac {y^{2}}{2}} x -{\mathrm e}^{\frac {y^{2}}{2}} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{\operatorname {RootOf}\left (c_{1}^{2} {\mathrm e}^{2 \textit {\_Z}}+2 x^{2} \textit {\_Z} +2 c_{1} {\mathrm e}^{\textit {\_Z}}+1\right )} c_{1} +1}{x} \]

Problem 11647

ODE	\[ \boxed {r^{\prime }+r \tan \left (t \right )=\cos \left (t \right )} \]
program solution	\[ r = \frac {t +c_{1}}{\sec \left (t \right )} \] Verified OK.
Maple solution	\[ r \left (t \right ) = \left (t +c_{1} \right ) \cos \left (t \right ) \]

Problem 11648

ODE	\[ \boxed {\cos \left (t \right ) r^{\prime }+\sin \left (t \right ) r=\cos \left (t \right )^{4}} \]
program solution	\[ r = \frac {\sin \left (t \right ) \cos \left (t \right )+2 c_{1} +t}{2 \sec \left (t \right )} \] Verified OK.
Maple solution	\[ r \left (t \right ) = \frac {\left (2 t +\sin \left (2 t \right )+4 c_{1} \right ) \cos \left (t \right )}{4} \]

Problem 11649

ODE	\[ \boxed {-y \cos \left (x \right )-\left (1+\sin \left (x \right )\right ) y^{\prime }=-\cos \left (x \right )^{2}} \]
program solution	\[ y = \frac {\cos \left (x \right ) \sin \left (x \right )-2 c_{1} +x}{2+2 \sin \left (x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (x \right ) \cos \left (x \right )+2 c_{1} +x}{2+2 \sin \left (x \right )} \]

Problem 11650

ODE	\[ \boxed {y \sin \left (2 x \right )+\left (1+\sin \left (x \right )^{2}\right ) y^{\prime }=\cos \left (x \right )} \]
program solution	\[ y = -\frac {\sin \left (x \right )+c_{1}}{\cos \left (x \right )^{2}-2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\sin \left (x \right )-c_{1}}{\cos \left (x \right )^{2}-2} \]

Problem 11651

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

Problem 11652

ODE	\[ \boxed {y^{\prime } x +y+2 x^{6} y^{4}=0} \]
program solution	\[ \frac {2 y^{3} x^{6}-1}{3 x^{3} y^{3}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1}{\left (2 x^{3}+c_{1} \right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {1+i \sqrt {3}}{2 \left (2 x^{3}+c_{1} \right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {i \sqrt {3}-1}{2 \left (2 x^{3}+c_{1} \right )^{\frac {1}{3}} x} \\ \end{align*}

Problem 11653

ODE	\[ \boxed {y^{\prime }+\left (4 y-\frac {8}{y^{3}}\right ) x=0} \]
program solution	\[ -\frac {x^{2}}{2}-\frac {\ln \left (y^{4}-2\right )}{16} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \left (2 \,{\mathrm e}^{8 x^{2}}+c_{1} \right )^{\frac {1}{4}} {\mathrm e}^{-2 x^{2}} \\ y \left (x \right ) &= -\left (2 \,{\mathrm e}^{8 x^{2}}+c_{1} \right )^{\frac {1}{4}} {\mathrm e}^{-2 x^{2}} \\ y \left (x \right ) &= -i \left (2 \,{\mathrm e}^{8 x^{2}}+c_{1} \right )^{\frac {1}{4}} {\mathrm e}^{-2 x^{2}} \\ y \left (x \right ) &= i \left (2 \,{\mathrm e}^{8 x^{2}}+c_{1} \right )^{\frac {1}{4}} {\mathrm e}^{-2 x^{2}} \\ \end{align*}

Problem 11654

ODE	\[ \boxed {x^{\prime }+\frac {\left (1+t \right ) x}{2 t}-\frac {1+t}{x t}=0} \]
program solution	\[ -t -\ln \left (t \right )-\ln \left (x^{2}-2\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} x \left (t \right ) &= \frac {\sqrt {t \,{\mathrm e}^{-t} c_{1} +2 t^{2}}}{t} \\ x \left (t \right ) &= -\frac {\sqrt {t \,{\mathrm e}^{-t} c_{1} +2 t^{2}}}{t} \\ \end{align*}

Problem 11655

ODE	\[ \boxed {y^{\prime } x -2 y=2 x^{4}} \] With initial conditions \begin {align*} [y \left (2\right ) = 8] \end {align*}
program solution	\[ y = x^{2} \left (x^{2}-2\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (x^{2}-2\right ) x^{2} \]

Problem 11656

ODE	\[ \boxed {y^{\prime }+3 x^{2} y=x^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {1}{3}+\frac {5 \,{\mathrm e}^{-x^{3}}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{3}+\frac {5 \,{\mathrm e}^{-x^{3}}}{3} \]

Problem 11657

ODE	\[ \boxed {{\mathrm e}^{x} \left (y-3 \left ({\mathrm e}^{x}+1\right )^{2}\right )+\left ({\mathrm e}^{x}+1\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{3 x}+3 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}+1}{{\mathrm e}^{x}+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+1 \]

Problem 11658

ODE	\[ \boxed {2 \left (y+1\right ) x -\left (x^{2}+1\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -5] \end {align*}
program solution	\[ y = -2 x^{2}-3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 x^{2}-3 \]

Problem 11659

ODE	\[ \boxed {r^{\prime }+r \tan \left (t \right )=\cos \left (t \right )^{2}} \] With initial conditions \begin {align*} \left [r \left (\frac {\pi }{4}\right ) = 1\right ] \end {align*}
program solution	\[ r = \sin \left (t \right ) \cos \left (t \right )+\frac {\cos \left (t \right ) \sqrt {2}}{2} \] Verified OK.
Maple solution	\[ r \left (t \right ) = \frac {\left (2 \sin \left (t \right )+\sqrt {2}\right ) \cos \left (t \right )}{2} \]

Problem 11660

ODE	\[ \boxed {-x+x^{\prime }=\sin \left (2 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {2 \,{\mathrm e}^{t}}{5}-\frac {\sin \left (2 t \right )}{5}-\frac {2 \cos \left (2 t \right )}{5} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {2 \cos \left (2 t \right )}{5}-\frac {\sin \left (2 t \right )}{5}+\frac {2 \,{\mathrm e}^{t}}{5} \]

Problem 11661

ODE	\[ \boxed {y^{\prime }+\frac {y}{2 x}-\frac {x}{y^{3}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ \frac {y^{4} x^{2}}{2}-\frac {x^{4}}{2} = {\frac {15}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {\frac {\sqrt {x^{4}+15}}{x}} \]

Problem 11662

ODE	\[ \boxed {y^{\prime } x +y-\left (x y\right )^{\frac {3}{2}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 4] \end {align*}
program solution	\[ y = \frac {4}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4}{x^{3}} \]

Problem 11663

ODE	\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} 2 & 0\le x <1 \\ 0 & 1\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \left \{\begin {array}{cc} 0 & x <0 \\ 2-2 \,{\mathrm e}^{-x} & x <1 \\ 2 \,{\mathrm e}^{1-x}-2 \,{\mathrm e}^{-x} & 1\le x \end {array}\right . \]

Problem 11664

ODE	\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} 5 & 0\le x <10 \\ 1 & 10\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 6] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \left \{\begin {array}{cc} 6 \,{\mathrm e}^{-x} & x <0 \\ {\mathrm e}^{-x}+5 & x <10 \\ {\mathrm e}^{-x}+1+4 \,{\mathrm e}^{10-x} & 10\le x \end {array}\right . \]

Problem 11665

ODE	\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} {\mathrm e}^{-x} & 0\le x <2 \\ {\mathrm e}^{-2} & 2\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \left \{\begin {array}{cc} {\mathrm e}^{-x} & x <0 \\ {\mathrm e}^{-x} \left (1+x \right ) & x <2 \\ 2 \,{\mathrm e}^{-x}+{\mathrm e}^{-2} & 2\le x \end {array}\right . \]

Problem 11666

ODE	\[ \boxed {\left (x +2\right ) y^{\prime }+y=\left \{\begin {array}{cc} 2 x & 0\le x <2 \\ 4 & 2\le x \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} \frac {8}{x +2} & x <0 \\ \frac {x^{2}+8}{x +2} & x <2 \\ \frac {4 x +4}{x +2} & 2\le x \\ 0 & \operatorname {otherwise} \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left \{\begin {array}{cc} 8 & x <0 \\ x^{2}+8 & x <2 \\ 4+4 x & 2\le x \end {array}\right .}{x +2} \]

Problem 11667

ODE	\[ \boxed {y^{\prime } a +y b=k \,{\mathrm e}^{-\lambda x}} \]
program solution	\[ y = -\frac {\left (k \,{\mathrm e}^{-\frac {x \left (a \lambda -b \right )}{a}} a -\lambda c_{1} a +c_{1} b \right ) {\mathrm e}^{-\frac {b x}{a}}}{\left (a \lambda -b \right ) a} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-k \,{\mathrm e}^{-\frac {x \left (a \lambda -b \right )}{a}}+c_{1} \left (a \lambda -b \right )\right ) {\mathrm e}^{-\frac {b x}{a}}}{a \lambda -b} \]

Problem 11668

ODE	\[ \boxed {y^{\prime }+y=2 \sin \left (x \right )+5 \sin \left (2 x \right )} \]
program solution	\[ y = {\mathrm e}^{-x} \left ({\mathrm e}^{x} \sin \left (2 x \right )+{\mathrm e}^{x} \sin \left (x \right )-\cos \left (x \right ) {\mathrm e}^{x}-2 \,{\mathrm e}^{x} \cos \left (2 x \right )+c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\cos \left (x \right )-2 \cos \left (2 x \right )+\sin \left (x \right )+\sin \left (2 x \right )+c_{1} {\mathrm e}^{-x} \]

Problem 11669

ODE	\[ \boxed {\cos \left (y\right ) y^{\prime }+\frac {\sin \left (y\right )}{x}=1} \]
program solution	\[ -\frac {x \left (-2 \sin \left (y\right )+x \right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\arcsin \left (\frac {-x^{2}+2 c_{1}}{2 x}\right ) \]

Problem 11670

ODE	\[ \boxed {\left (y+1\right ) y^{\prime }+x \left (y^{2}+2 y\right )=x} \]
program solution	\[ -\frac {x^{2}}{2}-\frac {\ln \left (y^{2}+2 y-1\right )}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -1-\sqrt {2+{\mathrm e}^{-x^{2}} c_{1}} \\ y \left (x \right ) &= -1+\sqrt {2+{\mathrm e}^{-x^{2}} c_{1}} \\ \end{align*}

Problem 11671

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

Problem 11672

ODE	\[ \boxed {y^{\prime }+y^{2}-x y=1} \]
program solution	\[ y = \frac {c_{3} \sqrt {2}\, {\mathrm e}^{-\frac {x^{2}}{2}}+x \sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )}{\sqrt {\pi }\, \left (c_{3} \operatorname {erf}\left (\frac {\sqrt {2}\, x}{2}\right )+1\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right ) x +2 c_{1} x +2 \,{\mathrm e}^{-\frac {x^{2}}{2}}}{\sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {x \sqrt {2}}{2}\right )+2 c_{1}} \]

Problem 11673

ODE	\[ \boxed {y^{\prime }+8 y^{2} x -4 x \left (4 x +1\right ) y=-8 x^{3}-4 x^{2}+1} \]
program solution	\[ y = \frac {2 \,{\mathrm e}^{\frac {8 x^{3}}{3}} c_{3} x +2 \,{\mathrm e}^{\frac {8}{3} x^{3}+2 x^{2}} x +{\mathrm e}^{\frac {8}{3} x^{3}+2 x^{2}}}{2 c_{3} {\mathrm e}^{\frac {8 x^{3}}{3}}+2 \,{\mathrm e}^{\frac {8}{3} x^{3}+2 x^{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (2 x +1\right ) {\mathrm e}^{\frac {8}{3} x^{3}+2 x^{2}}+2 \,{\mathrm e}^{\frac {8 x^{3}}{3}} x}{2 c_{1} {\mathrm e}^{\frac {8}{3} x^{3}+2 x^{2}}+2 \,{\mathrm e}^{\frac {8 x^{3}}{3}}} \]

Problem 11674

ODE	\[ \boxed {6 x^{2} y-\left (x^{3}+1\right ) y^{\prime }=0} \]
program solution	\[ y = {\mathrm e}^{6 c_{1}} \left (x^{3}+1\right )^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (x^{3}+1\right )^{2} \]

Problem 11675

ODE	\[ \boxed {\left (3 x^{2} y^{2}-x \right ) y^{\prime }+2 y^{3} x -y=0} \]
program solution	\[ y \left (y^{2} x -1\right ) x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {{\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} \left (2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}+4 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}\right )}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {{\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} \left (2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}+4 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}\right )}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {\left (i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {\left (i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {\left (-i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {\left (-i \left (-2 x^{2} 2^{\frac {1}{3}}+2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) \sqrt {3}-2 x^{2} 2^{\frac {1}{3}}+8 x {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}-2^{\frac {2}{3}} {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right )^{2} x^{4}\right )}^{\frac {1}{3}}\right ) {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}}}}{6 {\left (\left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2}-4 c_{1} x}+27 c_{1} -2 x \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}

Problem 11676

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

Problem 11677

ODE	\[ \boxed {y^{\prime } x -2 y=-x^{2}} \]
program solution	\[ y = -x^{2} \left (\ln \left (x \right )-c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-\ln \left (x \right )+c_{1} \right ) x^{2} \]

Problem 11678

ODE	\[ \boxed {-5 y+\left (y+x \right ) y^{\prime }=-3 x} \]
program solution	\[ -\ln \left (y-x \right )+2 \ln \left (-3 x +y\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {6 c_{1} x -\sqrt {8 c_{1} x +1}+1}{2 c_{1}} \\ y \left (x \right ) &= \frac {6 c_{1} x +1+\sqrt {8 c_{1} x +1}}{2 c_{1}} \\ \end{align*}

Problem 11679

ODE	\[ \boxed {{\mathrm e}^{2 x} y^{2}+\left (y \,{\mathrm e}^{2 x}-2 y\right ) y^{\prime }=0} \]
program solution	\[ y = {\mathrm e}^{-\frac {\ln \left ({\mathrm e}^{2 x}-2\right )}{2}-c_{1}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-\left ({\mathrm e}^{2 x}-2\right ) c_{1}}}{{\mathrm e}^{2 x}-2} \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-\left ({\mathrm e}^{2 x}-2\right ) c_{1}}}{{\mathrm e}^{2 x}-2} \\ \end{align*}

Problem 11680

ODE	\[ \boxed {8 y x^{3}+\left (x^{4}+1\right ) y^{\prime }=12 x^{3}} \]
program solution	\[ y = \frac {3 x^{8}+6 x^{4}+{\mathrm e}^{-8 c_{1}}+3}{2 x^{8}+4 x^{4}+2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3}{2}+\frac {c_{1}}{\left (x^{4}+1\right )^{2}} \]

Problem 11681

ODE	\[ \boxed {x y+y^{2}+2 y^{\prime } x^{2}=-2 x^{2}} \]
program solution	\[ y = \frac {\left (-2 c_{3} \sqrt {x}-x \right ) x}{c_{3} \sqrt {x}+x} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (c_{1} x -\sqrt {c_{1} x}-2\right ) x}{c_{1} x -1} \\ y \left (x \right ) &= -\frac {\left (c_{1} x +\sqrt {c_{1} x}-2\right ) x}{c_{1} x -1} \\ \end{align*}

Problem 11682

ODE	\[ \boxed {y^{\prime }-\frac {4 y^{2} x^{3}-3 x^{2} y}{x^{3}-2 y x^{4}}=0} \]
program solution	\[ -\frac {1}{\sqrt {x y-1}\, x \sqrt {x y}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x -\sqrt {x^{2}+4 c_{1}}}{2 x^{2}} \\ y \left (x \right ) &= \frac {x +\sqrt {x^{2}+4 c_{1}}}{2 x^{2}} \\ \end{align*}

Problem 11683

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

Problem 11684

ODE	\[ \boxed {y^{\prime }-\frac {2 x -7 y}{3 y-8 x}=0} \]
program solution	\[ -\frac {\ln \left (y-x \right )}{8}+\frac {\ln \left (2 x +3 y\right )}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-12 c_{1} x -\sqrt {-60 c_{1} x +1}+1}{18 c_{1}} \\ y \left (x \right ) &= \frac {-12 c_{1} x +1+\sqrt {-60 c_{1} x +1}}{18 c_{1}} \\ \end{align*}

Problem 11685

ODE	\[ \boxed {y^{\prime } x^{2}+x y-y^{3} x=0} \]
program solution	\[ -\ln \left (x \right )+\frac {\ln \left (y+1\right )}{2}-\ln \left (y\right )+\frac {\ln \left (y-1\right )}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {c_{1} x^{2}+1}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {c_{1} x^{2}+1}} \\ \end{align*}

Problem 11686

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

Problem 11687

ODE	\[ \boxed {y^{\prime }-\frac {2 x^{2}+y^{2}}{2 x y-x^{2}}=0} \]
program solution	\[ -2 x +\frac {y^{2}}{x}-y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {c_{1} x -\sqrt {9 c_{1}^{2} x^{2}+4 c_{1} x}}{2 c_{1}} \\ y \left (x \right ) &= \frac {c_{1} x +\sqrt {9 c_{1}^{2} x^{2}+4 c_{1} x}}{2 c_{1}} \\ \end{align*}

Problem 11688

ODE	\[ \boxed {y^{2}-2 y y^{\prime } x=-x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ x -\frac {y^{2}}{x} = -3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {\left (x +3\right ) x} \]

Problem 11689

ODE	\[ \boxed {2 y^{2}+\left (-x^{2}+1\right ) y y^{\prime }=-8} \] With initial conditions \begin {align*} [y \left (3\right ) = 0] \end {align*}
program solution	\[ \operatorname {arctanh}\left (x \right )+\frac {\ln \left (y^{2}+4\right )}{4} = \operatorname {arccoth}\left (3\right )-\frac {i \pi }{2}+\frac {\ln \left (2\right )}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {3 x^{2}-10 x +3}}{1+x} \\ y \left (x \right ) &= \frac {2 \sqrt {3 x^{2}-10 x +3}}{1+x} \\ \end{align*}

Problem 11690

ODE	\[ \boxed {{\mathrm e}^{2 x} y^{2}+{\mathrm e}^{2 x} y y^{\prime }=2 x} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ \frac {{\mathrm e}^{2 x} y^{2}}{2}-x^{2} = 2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x} \sqrt {2}\, \sqrt {{\mathrm e}^{2 x} \left (x^{2}+2\right )} \]

Problem 11691

ODE	\[ \boxed {2 y^{2} x +\left (2 x^{2} y+6 y^{2}\right ) y^{\prime }=-3 x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ x^{2} y^{2}+x^{3}+2 y^{3} = 21 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (1134-54 x^{3}-x^{6}+6 \sqrt {3 x^{9}+18 x^{6}-3402 x^{3}+35721}\right )^{\frac {1}{3}}}{6}+\frac {x^{4}}{6 \left (1134-54 x^{3}-x^{6}+6 \sqrt {3 x^{9}+18 x^{6}-3402 x^{3}+35721}\right )^{\frac {1}{3}}}-\frac {x^{2}}{6} \]

Problem 11692

ODE	\[ \boxed {4 y y^{\prime } x -y^{2}=1} \] With initial conditions \begin {align*} [y \left (2\right ) = 1] \end {align*}
program solution	\[ -\ln \left (x \right )+2 \ln \left (1+y^{2}\right ) = \ln \left (2\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {\sqrt {2}\, \sqrt {x}-1} \]

Problem 11693

ODE	\[ \boxed {y^{\prime }-\frac {2 x +7 y}{-2 y+2 x}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ -\ln \left (x +2 y\right )+2 \ln \left (2 x +y\right ) = -\ln \left (5\right )+4 \ln \left (2\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 \sqrt {16-15 x}}{5}-2 x +\frac {16}{5} \]

Problem 11694

ODE	\[ \boxed {y^{\prime }-\frac {x y}{x^{2}+1}=0} \] With initial conditions \begin {align*} \left [y \left (\sqrt {15}\right ) = 2\right ] \end {align*}
program solution	\[ y = \frac {\sqrt {x^{2}+1}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {x^{2}+1}}{2} \]

Problem 11695

ODE	\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} 1 & 0\le x <2 \\ 0 & 0
program solution	N/A
Maple solution	\[ y \left (x \right ) = \left \{\begin {array}{cc} 0 & x <0 \\ 1-{\mathrm e}^{-x} & x <2 \\ {\mathrm e}^{2-x}-{\mathrm e}^{-x} & 2\le x \end {array}\right . \]

Problem 11696

ODE	\[ \boxed {\left (x +2\right ) y^{\prime }+y=\left \{\begin {array}{cc} 2 x & 0\le x \le 2 \\ 4 & 2
program solution	\[ y = \left \{\begin {array}{cc} \frac {8}{x +2} & x <0 \\ \frac {x^{2}+8}{x +2} & x <2 \\ \frac {4 x +4}{x +2} & 2\le x \\ 0 & \operatorname {otherwise} \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left \{\begin {array}{cc} 8 & x <0 \\ x^{2}+8 & x <2 \\ 4+4 x & 2\le x \end {array}\right .}{x +2} \]

Problem 11697

ODE	\[ \boxed {y^{\prime } x^{2}+x y-\frac {y^{3}}{x}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ \frac {1}{y^{2}} = \frac {x^{4}+1}{2 x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 x}{\sqrt {2 x^{4}+2}} \]

Problem 11698

ODE	\[ \boxed {5 x y+4 y^{2}+\left (2 x y+x^{2}\right ) y^{\prime }=-1} \]
program solution	\[ x^{5} y+y^{2} x^{4}+\frac {x^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-x^{3}-\sqrt {x^{6}-x^{4}-4 c_{1}}}{2 x^{2}} \\ y \left (x \right ) &= \frac {-x^{3}+\sqrt {x^{6}-x^{4}-4 c_{1}}}{2 x^{2}} \\ \end{align*}

Problem 11699

ODE	\[ \boxed {\tan \left (y\right )+\left (x -x^{2} \tan \left (y\right )\right ) y^{\prime }=-2 x} \]
program solution	\[ \left (\cos \left (y\right ) x +\sin \left (y\right )\right ) x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \arctan \left (\frac {-\sqrt {x^{4}-c_{1}^{2}+x^{2}}\, x -c_{1}}{\left (x^{2}+1\right ) x}, \frac {-c_{1} x +\sqrt {x^{4}-c_{1}^{2}+x^{2}}}{\left (x^{2}+1\right ) x}\right ) \\ y \left (x \right ) &= \arctan \left (\frac {\sqrt {x^{4}-c_{1}^{2}+x^{2}}\, x -c_{1}}{\left (x^{2}+1\right ) x}, \frac {-c_{1} x -\sqrt {x^{4}-c_{1}^{2}+x^{2}}}{\left (x^{2}+1\right ) x}\right ) \\ \end{align*}

Problem 11700

ODE	\[ \boxed {\left (x +1\right ) y^{2}+y+\left (2 x y+1\right ) y^{\prime }=0} \]
program solution	\[ \left (x y+1\right ) y \,{\mathrm e}^{x} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-1+\sqrt {{\mathrm e}^{x} \left (-4 c_{1} x +{\mathrm e}^{x}\right )}\, {\mathrm e}^{-x}}{2 x} \\ y \left (x \right ) &= \frac {-\sqrt {{\mathrm e}^{x} \left (-4 c_{1} x +{\mathrm e}^{x}\right )}\, {\mathrm e}^{-x}-1}{2 x} \\ \end{align*}