2.17.127 Problems 12601 to 12700

Problem 12601

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

Problem 12602

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

Problem 12603

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

Problem 12604

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 3, y^{\prime }\left (1\right ) = -1] \end {align*}
program solution	\[ y = \frac {2 \,{\mathrm e}^{2 x} {\mathrm e}^{-2}}{3}+\frac {7 \,{\mathrm e}^{-x} {\mathrm e}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {7 \,{\mathrm e}^{1-x}}{3}+\frac {2 \,{\mathrm e}^{2 x -2}}{3} \]

Problem 12605

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y \left (2\right ) = 0] \end {align*}
program solution	\[ y = \frac {-{\mathrm e}^{2 x +6}-{\mathrm e}^{-x +6}+{\mathrm e}^{12-x}+{\mathrm e}^{2 x}}{{\mathrm e}^{12}-2 \,{\mathrm e}^{6}+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{6-x}-{\mathrm e}^{2 x}}{{\mathrm e}^{6}-1} \]

Problem 12606

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (2\right ) = 1] \end {align*}
program solution	\[ y = \frac {2 \,{\mathrm e}^{2 x +8}-{\mathrm e}^{2-x}-2 \,{\mathrm e}^{8-x}+{\mathrm e}^{2 x +2}}{4 \,{\mathrm e}^{12}+4 \,{\mathrm e}^{6}+1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{2-x} \left ({\mathrm e}^{3 x}-1\right )}{2 \,{\mathrm e}^{6}+1} \]

Problem 12607

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

Problem 12608

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

Problem 12609

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

Problem 12610

ODE	\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (2\right ) = -12] \end {align*}
program solution	\[ y = -2 x^{3}+3 x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 x^{3}+3 x^{2} \]

Problem 12611

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

Problem 12612

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

Problem 12613

ODE	\[ \boxed {x^{2} y^{\prime \prime }-4 y^{\prime } x +6 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (2\right ) = -1] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 12614

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

Problem 12615

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

Problem 12616

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

Problem 12617

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

Problem 12618

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

Problem 12619

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

Problem 12620

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

Problem 12621

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

Problem 12622

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

Problem 12623

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

Problem 12624

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

Problem 12625

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

Problem 12626

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

Problem 12627

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

Problem 12628

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

Problem 12629

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

Problem 12630

ODE	\[ \boxed {y^{\prime }-y^{3}=x^{3}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 12631

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

Problem 12632

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

Problem 12633

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

Problem 12634

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

Problem 12635

ODE	\[ \boxed {y^{\prime }-\frac {1}{\sqrt {15-x^{2}-y^{2}}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 12636

ODE	\[ \boxed {y^{\prime }-\frac {3 y}{\left (x -5\right ) \left (3+x \right )}={\mathrm e}^{-x}} \]
program solution	\[ \int _{}^{x}-\frac {\left (\textit {\_a}^{2}-2 \textit {\_a} -15\right ) {\mathrm e}^{-\textit {\_a}}+3 y}{\left (\textit {\_a} -5\right )^{\frac {11}{8}} \left (3+\textit {\_a} \right )^{\frac {5}{8}}}d \textit {\_a} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\int \frac {{\mathrm e}^{-x} \left (x +3\right )^{\frac {3}{8}}}{\left (x -5\right )^{\frac {3}{8}}}d x +c_{1} \right ) \left (x -5\right )^{\frac {3}{8}}}{\left (x +3\right )^{\frac {3}{8}}} \]

Problem 12637

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

Problem 12638

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

Problem 12639

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

Problem 12640

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

Problem 12641

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

Problem 12642

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

Problem 12643

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

Problem 12644

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

Problem 12645

ODE	\[ \boxed {y^{\prime }-\left (y x \right )^{\frac {1}{3}}=0} \]
program solution	\[ -\frac {3 x \left (y x \right )^{\frac {1}{3}}}{4 y^{\frac {1}{3}}}+\frac {3 y^{\frac {2}{3}}}{2} = c_{1} \] Verified OK.
Maple solution	\[ -\frac {\left (\left (-4 x^{5} c_{1} +32 y \left (x \right )^{2} c_{1} x +2 x \right ) \left (y \left (x \right ) x \right )^{\frac {2}{3}}+\left (x^{3}+4 \left (y \left (x \right ) x \right )^{\frac {1}{3}} y \left (x \right )\right ) \left (x^{4} c_{1} -8 c_{1} y \left (x \right )^{2}+1\right )\right ) x}{\left (x^{4}-8 y \left (x \right )^{2}\right ) \left (-2 \left (y \left (x \right ) x \right )^{\frac {2}{3}}+x^{2}\right )^{2}} = 0 \]

Problem 12646

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

Problem 12647

ODE	\[ \boxed {y^{\prime }+\frac {y}{x}-y^{\frac {1}{4}}=0} \]
program solution	\[ \frac {4 \left (y x \right )^{\frac {3}{4}} \left (7 y^{\frac {3}{4}}-3 x \right )}{21 y^{\frac {3}{4}}} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right )^{\frac {3}{4}}-\frac {3 x}{7}-\frac {c_{1}}{x^{\frac {3}{4}}} = 0 \]

Problem 12648

ODE	\[ \boxed {y^{\prime }-4 y=-5} \] With initial conditions \begin {align*} [y \left (1\right ) = 4] \end {align*}
program solution	\[ -\frac {\ln \left (2\right )}{2}+\frac {\ln \left (4 y-5\right )}{4} = x -1+\frac {\ln \left (11\right )}{4}-\frac {\ln \left (2\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {5}{4}+\frac {11 \,{\mathrm e}^{-4+4 x}}{4} \]

Problem 12649

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

Problem 12650

ODE	\[ \boxed {y^{\prime }-a y=b} \] With initial conditions \begin {align*} [y \left (c \right ) = d] \end {align*}
program solution	\[ \frac {\ln \left (a y+b \right )}{a} = \frac {\ln \left (a d +b \right )+\left (-c +x \right ) a}{a} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (a d +b \right ) {\mathrm e}^{-a \left (c -x \right )}-b}{a} \]

Problem 12651

ODE	\[ \boxed {y^{\prime }=x^{2}+{\mathrm e}^{x}-\sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (2\right ) = -1] \end {align*}
program solution	\[ y = \frac {x^{3}}{3}+\cos \left (x \right )+{\mathrm e}^{x}-\frac {11}{3}-\cos \left (2\right )-{\mathrm e}^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{3}}{3}+\cos \left (x \right )+{\mathrm e}^{x}-\frac {11}{3}-\cos \left (2\right )-{\mathrm e}^{2} \]

Problem 12652

ODE	\[ \boxed {y^{\prime }-y x=\frac {1}{x^{2}+1}} \] With initial conditions \begin {align*} [y \left (-5\right ) = 0] \end {align*}
program solution	\[ y = -\frac {\int _{-5}^{x}\frac {{\mathrm e}^{-\frac {\textit {\_a}^{2}}{2}}}{\textit {\_a}^{2}+1}d \textit {\_a}}{-{\mathrm e}^{-\frac {x^{2}}{2}}+\int _{-5}^{x}\frac {{\mathrm e}^{-\frac {\textit {\_a}^{2}}{2}} \textit {\_a}}{\textit {\_a}^{2}+1}d \textit {\_a} +\int _{-5}^{x}\frac {{\mathrm e}^{-\frac {\textit {\_a}^{2}}{2}} \textit {\_a}^{3}}{\textit {\_a}^{2}+1}d \textit {\_a} -\left (\int _{-5}^{x}\textit {\_a} \,{\mathrm e}^{-\frac {\textit {\_a}^{2}}{2}}d \textit {\_a} \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\int _{-5}^{x}\frac {{\mathrm e}^{-\frac {\textit {\_z1}^{2}}{2}}}{\textit {\_z1}^{2}+1}d \textit {\_z1} \right ) {\mathrm e}^{\frac {x^{2}}{2}} \]

Problem 12653

ODE	\[ \boxed {y^{\prime }-\frac {y}{x}=\cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (-1\right ) = 0] \end {align*}
program solution	\[ y = -i \pi x +x \,\operatorname {Ci}\left (x \right )-\operatorname {Ci}\left (1\right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {Ci}\left (x \right )-\operatorname {Ci}\left (1\right )-i \pi \right ) x \]

Problem 12654

ODE	\[ \boxed {y^{\prime }-\frac {y}{x}=\tan \left (x \right )} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 0] \end {align*}
program solution	\[ y = \left (\int _{\pi }^{x}\frac {\tan \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} \right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\int _{\pi }^{x}\frac {\tan \left (\textit {\_z1} \right )}{\textit {\_z1}}d \textit {\_z1} \right ) x \]

Problem 12655

ODE	\[ \boxed {y^{\prime }-\frac {y}{-x^{2}+4}=\sqrt {x}} \] With initial conditions \begin {align*} [y \left (3\right ) = 4] \end {align*}
program solution	\[ y = \frac {4 \sqrt {\frac {x +2}{\sqrt {-x^{2}+4}}}\, 5^{\frac {3}{4}}+5 \sqrt {\frac {x +2}{\sqrt {-x^{2}+4}}}\, \left (\int _{3}^{x}\frac {\sqrt {\textit {\_a}}}{\sqrt {\frac {\textit {\_a} +2}{\sqrt {-\textit {\_a}^{2}+4}}}}d \textit {\_a} \right ) \sqrt {-i}}{5 \sqrt {-i}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (4 \,5^{\frac {3}{4}}+5 \left (\int _{3}^{x}\frac {\sqrt {\textit {\_z1}}\, \left (\textit {\_z1} -2\right )^{\frac {1}{4}}}{\left (2+\textit {\_z1} \right )^{\frac {1}{4}}}d \textit {\_z1} \right )\right ) \left (x +2\right )^{\frac {1}{4}}}{5 \left (x -2\right )^{\frac {1}{4}}} \]

Problem 12656

ODE	\[ \boxed {y^{\prime }-\frac {y}{-x^{2}+4}=\sqrt {x}} \] With initial conditions \begin {align*} [y \left (1\right ) = -3] \end {align*}
program solution	\[ y = -3^{\frac {3}{4}} \sqrt {\frac {x +2}{\sqrt {-x^{2}+4}}}+\left (\int _{1}^{x}\frac {\sqrt {\textit {\_a}}}{\sqrt {\frac {\textit {\_a} +2}{\sqrt {-\textit {\_a}^{2}+4}}}}d \textit {\_a} \right ) \sqrt {\frac {x +2}{\sqrt {-x^{2}+4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (x +2\right )^{\frac {1}{4}} \left (-2 \left (\int _{1}^{x}\frac {\sqrt {\textit {\_z1}}\, \left (\textit {\_z1} -2\right )^{\frac {1}{4}}}{\left (2+\textit {\_z1} \right )^{\frac {1}{4}}}d \textit {\_z1} \right )+\left (1+i\right ) \sqrt {2}\, 3^{\frac {3}{4}}\right )}{2 \left (x -2\right )^{\frac {1}{4}}} \]

Problem 12657

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

Problem 12658

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

Problem 12659

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

Problem 12660

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

Problem 12661

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

Problem 12662

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

Problem 12663

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

Problem 12664

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

Problem 12665

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

Problem 12666

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

Problem 12667

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

Problem 12668

ODE	\[ \boxed {y^{\prime }=\tan \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = -\ln \left (\cos \left (x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\ln \left (\cos \left (x \right )\right ) \]

Problem 12669

ODE	\[ \boxed {y^{\prime }=\tan \left (x \right )} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 0] \end {align*}
program solution	\[ y = -\ln \left (\cos \left (x \right )\right )+i \pi \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\ln \left (\cos \left (x \right )\right )+i \pi \]

Problem 12670

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

Problem 12671

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

Problem 12672

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

Problem 12673

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

Problem 12674

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

Problem 12675

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

Problem 12676

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

Problem 12677

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

Problem 12678

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

Problem 12679

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

Problem 12680

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

Problem 12681

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

Problem 12682

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

Problem 12683

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

Problem 12684

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

Problem 12685

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

Problem 12686

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

Problem 12687

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

Problem 12688

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

Problem 12689

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

Problem 12690

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

Problem 12691

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

Problem 12692

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

Problem 12693

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

Problem 12694

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

Problem 12695

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

Problem 12696

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

Problem 12697

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

Problem 12698

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

Problem 12699

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

Problem 12700

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