2.17.136 Problems 13501 to 13600

Problem 13501

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

Problem 13502

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

Problem 13503

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

Problem 13504

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }=9 \,{\mathrm e}^{-3 x}} \]
program solution	\[ y = \frac {c_{1}}{4}-3 \,{\mathrm e}^{-3 x}+c_{2} {\mathrm e}^{-4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-4 x} c_{1}}{4}-3 \,{\mathrm e}^{-3 x}+c_{2} \]

Problem 13505

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

Problem 13506

ODE	\[ \boxed {x y^{\prime \prime }+4 y^{\prime }=18 x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 8, y^{\prime }\left (1\right ) = -3] \end {align*}
program solution	\[ y = \frac {x^{6}+5 x^{3}+2}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3}+\frac {2}{x^{3}}+5 \]

Problem 13507

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

Problem 13508

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 8, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = 5 \,{\mathrm e}^{x}+3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 3+5 \,{\mathrm e}^{x} \]

Problem 13509

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

Problem 13510

ODE	\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 10, y^{\prime }\left (0\right ) = 5, y^{\prime \prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = 8+2 \,{\mathrm e}^{x}+3 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = 8+3 x +2 \,{\mathrm e}^{x} \]

Problem 13511

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

Problem 13512

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

Problem 13513

ODE	\[ \boxed {2 x y^{\prime } y^{\prime \prime }-{y^{\prime }}^{2}=-1} \] With initial conditions \begin {align*} \left [y \left (1\right ) = 0, y^{\prime }\left (1\right ) = \sqrt {3}\right ] \end {align*}
program solution	\[ y = \frac {\left (2 x +1\right )^{\frac {3}{2}}}{3}-\sqrt {3} \] Verified OK. \[ y = -\frac {\left (2 x +1\right )^{\frac {3}{2}}}{3}+\sqrt {3} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \frac {\left (2 x +1\right )^{\frac {3}{2}}}{3}-\sqrt {3} \]

Problem 13514

ODE	\[ \boxed {3 y y^{\prime \prime }-2 {y^{\prime }}^{2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = 9] \end {align*}
program solution	\[ y = 27 x^{3}-54 x^{2}+36 x -8 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (3 x -2\right )^{3} \]

Problem 13515

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

Problem 13516

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

Problem 13517

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

Problem 13518

ODE	\[ \boxed {y^{\prime \prime }+2 x {y^{\prime }}^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = 3 \]

Problem 13519

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

Problem 13520

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

Problem 13521

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*}
program solution	\[ y = \tan \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (x \right ) \]

Problem 13522

ODE	\[ \boxed {y^{\prime \prime }-2 y y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = -\frac {1}{-1+x} \]

Problem 13523

ODE	\[ \boxed {y^{\prime \prime }-2 y y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = 1 \]

Problem 13524

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*}
program solution	\[ y = -\tanh \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\tanh \left (x \right ) \]

Problem 13525

ODE	\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }-4 y=x^{3}} \]
program solution
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-\frac {x^{3}}{3}} \left (\int \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right ) x^{3}d x +\left (-c_{1} -\left (\int \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) x^{3}d x \right )\right ) \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right )-c_{2} \right ) \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \]

Problem 13526

ODE	\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }-4 y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{3}}{3}} \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right )\right ) \]

Problem 13527

ODE	\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }-4 y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x^{3}}{3}} \operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right ) \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{\frac {x^{3}}{3}}}{\operatorname {HeunT}\left (-4 \,3^{\frac {2}{3}}, -3, 0, \frac {3^{\frac {2}{3}} x}{3}\right )^{2}}d x \right )\right ) \]

Problem 13528

ODE	\[ \boxed {y^{\prime \prime }+x^{2} y^{\prime }+4 y-y^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 13529

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

Problem 13530

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

Problem 13531

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

Problem 13532

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=30 \,{\mathrm e}^{3 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right ) {\mathrm e}^{x}}{2}+\frac {15 \,{\mathrm e}^{3 x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \sin \left (2 x \right ) c_{2} +{\mathrm e}^{x} \cos \left (2 x \right ) c_{1} +\frac {15 \,{\mathrm e}^{3 x}}{4} \]

Problem 13533

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+6 y^{\prime \prime }+3 y^{\prime }-83 y=25} \]
program solution	\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =3\right ) x} c_{1} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =1\right ) x} c_{2} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =4\right ) x} c_{3} +{\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =2\right ) x} c_{4} -\frac {25}{83} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {25}{83}+c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =1\right ) x}+c_{2} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =2\right ) x}+c_{3} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =3\right ) x}+c_{4} {\mathrm e}^{\operatorname {RootOf}\left (\textit {\_Z}^{4}+6 \textit {\_Z}^{2}+3 \textit {\_Z} -83, \operatorname {index} =4\right ) x} \]

Problem 13534

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

Problem 13535

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{2 x} {\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{3 x}+c_{2} {\mathrm e}^{2 x} \]

Problem 13536

ODE	\[ \boxed {y^{\prime \prime }-10 y^{\prime }+25 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{5 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{5 x} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{5 x} \left (c_{2} x +c_{1} \right ) \]

Problem 13537

ODE	\[ \boxed {x^{2} y^{\prime \prime }-6 y^{\prime } x +12 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{3} \end {align*}
program solution	\[ y = c_{2} x^{4}+c_{1} x^{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{1} x +c_{2} \right ) \]

Problem 13538

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x -2 \sqrt {x}\, c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}+c_{2} x \]

Problem 13539

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sqrt {x} \end {align*}
program solution	\[ y = c_{1} \sqrt {x}+c_{2} \ln \left (x \right ) \sqrt {x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} +c_{2} \ln \left (x \right )\right ) \sqrt {x} \]

Problem 13540

ODE	\[ \boxed {y^{\prime \prime }-\left (4+\frac {2}{x}\right ) y^{\prime }+\left (4+\frac {4}{x}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{2 x} x^{3}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{2} x^{3}+c_{1} \right ) \]

Problem 13541

ODE	\[ \boxed {\left (1+x \right ) y^{\prime \prime }+y^{\prime } x -y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x} \end {align*}
program solution	\[ y = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{-x} x \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +c_{2} {\mathrm e}^{-x} \]

Problem 13542

ODE	\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{x}-4 y x^{2}=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x^{2}} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{-x^{2}}+\frac {c_{2} {\mathrm e}^{-x^{2}} {\mathrm e}^{2 x^{2}}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sinh \left (x^{2}\right )+c_{2} \cosh \left (x^{2}\right ) \]

Problem 13543

ODE	\[ \boxed {y^{\prime \prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x \right ) \end {align*}
program solution	\[ y = c_{1} \sin \left (x \right )-c_{2} \sin \left (x \right ) \cot \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right ) \]

Problem 13544

ODE	\[ \boxed {x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{x} \end {align*}
program solution	\[ y = \frac {c_{1}}{x}-\frac {c_{2} {\mathrm e}^{-2 x}}{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} +{\mathrm e}^{-2 x} c_{2}}{x} \]

Problem 13545

ODE	\[ \boxed {\sin \left (x \right )^{2} y^{\prime \prime }-2 \cos \left (x \right ) \sin \left (x \right ) y^{\prime }+\left (1+\cos \left (x \right )^{2}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x \right ) \end {align*}
program solution	\[ y = c_{1} \sin \left (x \right )+c_{2} \sin \left (x \right ) x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (x \right ) \left (c_{2} x +c_{1} \right ) \]

Problem 13546

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +\left (x^{2}+2\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (x \right ) x \end {align*}
program solution	\[ y = \sin \left (x \right ) x c_{1} -c_{2} \sin \left (x \right ) x \cot \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right )\right ) \]

Problem 13547

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {x}{2}-\frac {1}{2 x} \end {align*}
program solution	\[ y = \left (\frac {x}{2}-\frac {1}{2 x}\right ) c_{1} -\frac {2 c_{2} \left (\frac {x}{2}-\frac {1}{2 x}\right )}{x^{2}-1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\ln \left (x \right )\right )+c_{2} \cos \left (\ln \left (x \right )\right ) \]

Problem 13548

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (x^{2}-\frac {1}{4}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\cos \left (x \right )}{\sqrt {x}} \end {align*}
program solution	\[ y = \frac {\cos \left (x \right ) c_{1}}{\sqrt {x}}+\frac {c_{2} \cos \left (x \right ) \tan \left (x \right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right )}{\sqrt {x}} \]

Problem 13549

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=9 \,{\mathrm e}^{2 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{3 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{3 x}-\frac {c_{2} {\mathrm e}^{x}}{2}-9 \,{\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{3 x}+c_{1} {\mathrm e}^{x}-9 \,{\mathrm e}^{2 x} \]

Problem 13550

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+8 y={\mathrm e}^{4 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 x} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{4 x}}{2}+\frac {{\mathrm e}^{4 x} \left (2 x -1\right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (x +c_{1} -\frac {1}{2}\right ) {\mathrm e}^{2 x}+2 c_{2} \right ) {\mathrm e}^{2 x}}{2} \]

Problem 13551

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -y=\sqrt {x}} \] Given that one solution of the ode is \begin {align*} y_1 &= x \end {align*}
program solution	\[ y = c_{1} x -\frac {c_{2}}{2 x}-\frac {4 \sqrt {x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 c_{2} x^{2}-4 x^{\frac {3}{2}}+3 c_{1}}{3 x} \]

Problem 13552

ODE	\[ \boxed {x^{2} y^{\prime \prime }-20 y=27 x^{5}} \] Given that one solution of the ode is \begin {align*} y_1 &= x^{5} \end {align*}
program solution	\[ y = x^{5} c_{1} -\frac {c_{2}}{9 x^{4}}+x^{5} \left (-\frac {1}{3}+3 \ln \left (x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {9 x^{9} \ln \left (x \right )+\left (3 c_{2} -1\right ) x^{9}+3 c_{1}}{3 x^{4}} \]

Problem 13553

ODE	\[ \boxed {x y^{\prime \prime }+\left (2 x +2\right ) y^{\prime }+2 y=8 \,{\mathrm e}^{2 x}} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{x} \end {align*}
program solution	\[ y = \frac {c_{1}}{x}-\frac {c_{2} {\mathrm e}^{-2 x}}{2 x}+\frac {{\mathrm e}^{2 x}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-2 x} c_{2} +{\mathrm e}^{2 x}+c_{1}}{x} \]

Problem 13554

ODE	\[ \boxed {\left (1+x \right ) y^{\prime \prime }+y^{\prime } x -y=\left (1+x \right )^{2}} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-x} \end {align*}
program solution	\[ y = {\mathrm e}^{-x} c_{1} +c_{2} x +x^{2}-x +1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} x +c_{1} {\mathrm e}^{-x}+x^{2}+1 \]

Problem 13555

ODE	\[ \boxed {y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x}+x \,{\mathrm e}^{3 x} c_{2} +x^{2} {\mathrm e}^{3 x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (x^{2} c_{3} +c_{2} x +c_{1} \right ) \]

Problem 13556

ODE	\[ \boxed {y^{\prime \prime \prime }-9 y^{\prime \prime }+27 y^{\prime }-27 y={\mathrm e}^{3 x} \sin \left (x \right )} \]
program solution	\[ y = {\mathrm e}^{3 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+{\mathrm e}^{3 x} \cos \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (\cos \left (x \right )+c_{1} +c_{2} x +x^{2} c_{3} \right ) \]

Problem 13557

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+16 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{2} +x^{2} {\mathrm e}^{2 x} c_{3} +x^{3} {\mathrm e}^{2 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{4} x^{3}+x^{2} c_{3} +c_{2} x +c_{1} \right ) \]

Problem 13558

ODE	\[ \boxed {x^{3} y^{\prime \prime \prime }-4 y^{\prime \prime }+10 y^{\prime }-12 y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 13559

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

Problem 13560

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

Problem 13561

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

Problem 13562

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

Problem 13563

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^{\prime }\left (1\right ) = 4] \end {align*}
program solution	\[ y = 4 x^{3}-4 x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4 x^{3}-4 x^{2} \]

Problem 13564

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

Problem 13565

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

Problem 13566

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

Problem 13567

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

Problem 13568

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^{\prime }\left (0\right ) = -4] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 13569

ODE	\[ \boxed {x y^{\prime \prime }-y^{\prime }+4 y x^{3}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 13570

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

Problem 13571

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

Problem 13572

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

Problem 13573

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

Problem 13574

ODE	\[ \boxed {y^{\prime \prime }-10 y^{\prime }+9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 8, y^{\prime }\left (0\right ) = -24] \end {align*}
program solution	\[ y = -4 \,{\mathrm e}^{9 x}+12 \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 12 \,{\mathrm e}^{x}-4 \,{\mathrm e}^{9 x} \]

Problem 13575

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = 1 \]

Problem 13576

ODE	\[ \boxed {y^{\prime \prime \prime }-9 y^{\prime }=0} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{-3 x}+c_{3} {\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +{\mathrm e}^{-3 x} c_{2} +c_{3} {\mathrm e}^{3 x} \]

Problem 13577

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-10 y^{\prime \prime }+9 y=0} \]
program solution	\[ y = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{x}+{\mathrm e}^{-3 x} c_{3} +{\mathrm e}^{3 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{6 x} c_{2} +c_{4} {\mathrm e}^{4 x}+c_{3} {\mathrm e}^{2 x}+c_{1} \right ) {\mathrm e}^{-3 x} \]

Problem 13578

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+10 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{5 x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{2 x} \]

Problem 13579

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-24 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-6 x}+\frac {c_{2} {\mathrm e}^{4 x}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{10 x} c_{1} +c_{2} \right ) {\mathrm e}^{-6 x} \]

Problem 13580

ODE	\[ \boxed {y^{\prime \prime }-25 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 x}+\frac {c_{2} {\mathrm e}^{5 x}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{-5 x} \]

Problem 13581

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

Problem 13582

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

Problem 13583

ODE	\[ \boxed {3 y^{\prime \prime }+7 y^{\prime }-6 y=0} \]
program solution	\[ y = {\mathrm e}^{-3 x} c_{1} +\frac {3 c_{2} {\mathrm e}^{\frac {2 x}{3}}}{11} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{\frac {11 x}{3}}+c_{1} \right ) {\mathrm e}^{-3 x} \]

Problem 13584

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+15 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {3 \,{\mathrm e}^{5 x}}{2}+\frac {5 \,{\mathrm e}^{3 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3 \,{\mathrm e}^{5 x}}{2}+\frac {5 \,{\mathrm e}^{3 x}}{2} \]

Problem 13585

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+15 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{5 x}}{2}-\frac {{\mathrm e}^{3 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{5 x}}{2}-\frac {{\mathrm e}^{3 x}}{2} \]

Problem 13586

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+15 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = 19] \end {align*}
program solution	\[ y = 2 \,{\mathrm e}^{5 x}+3 \,{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{5 x}+3 \,{\mathrm e}^{3 x} \]

Problem 13587

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

Problem 13588

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

Problem 13589

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

Problem 13590

ODE	\[ \boxed {y^{\prime \prime }-10 y^{\prime }+25 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{5 x} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{5 x} \left (c_{2} x +c_{1} \right ) \]

Problem 13591

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

Problem 13592

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

Problem 13593

ODE	\[ \boxed {25 y^{\prime \prime }-10 y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x}{5}}+c_{2} x \,{\mathrm e}^{\frac {x}{5}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x}{5}} \left (c_{2} x +c_{1} \right ) \]

Problem 13594

ODE	\[ \boxed {16 y^{\prime \prime }-24 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {3 x}{4}}+c_{2} x \,{\mathrm e}^{\frac {3 x}{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {3 x}{4}} \left (c_{2} x +c_{1} \right ) \]

Problem 13595

ODE	\[ \boxed {9 y^{\prime \prime }+12 y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {2 x}{3}}+c_{2} x \,{\mathrm e}^{-\frac {2 x}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {2 x}{3}} \left (c_{2} x +c_{1} \right ) \]

Problem 13596

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = {\mathrm e}^{4 x} \left (1-4 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{4 x} \left (-4 x +1\right ) \]

Problem 13597

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = x \,{\mathrm e}^{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{4 x} x \]

Problem 13598

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+16 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 14] \end {align*}
program solution	\[ y = {\mathrm e}^{4 x} \left (2 x +3\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{4 x} \left (3+2 x \right ) \]

Problem 13599

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

Problem 13600

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