2.17.155 Problems 15401 to 15500

Problem 15401

ODE	\[ \boxed {x^{2} y^{\prime \prime }+4 x y^{\prime }+2 y=2 \ln \left (x \right )^{2}+12 x} \]
program solution	\[ y = \frac {x^{2} \ln \left (x \right )^{2}-3 \ln \left (x \right ) x^{2}+\frac {7 x^{2}}{2}+2 x^{3}+c_{1} x +c_{2}}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2}}{x^{2}}+2 x +\frac {7}{2}+\frac {c_{1}}{x}-3 \ln \left (x \right )+\ln \left (x \right )^{2} \]

Problem 15402

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

Problem 15403

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

Problem 15404

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

Problem 15405

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

Problem 15406

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

Problem 15407

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

Problem 15408

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

Problem 15409

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

Problem 15410

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

Problem 15411

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

Problem 15412

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

Problem 15413

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

Problem 15414

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

Problem 15415

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

Problem 15416

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

Problem 15417

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

Problem 15418

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

Problem 15419

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

Problem 15420

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

Problem 15421

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

Problem 15422

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

Problem 15423

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

Problem 15424

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

Problem 15425

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

Problem 15426

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

Problem 15427

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

Problem 15428

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

Problem 15429

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

Problem 15430

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

Problem 15431

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

Problem 15432

ODE	\[ \boxed {4 x y^{\prime \prime }+2 y^{\prime }+y=\frac {6+x}{x^{2}}} \] With initial conditions \begin {align*} [y \left (\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \textit {undefined} \]

Problem 15433

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

Problem 15434

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

Problem 15435

ODE	\[ \boxed {2 x^{2} \left (2-\ln \left (x \right )\right ) y^{\prime \prime }+x \left (4-\ln \left (x \right )\right ) y^{\prime }-y=\frac {\left (2-\ln \left (x \right )\right )^{2}}{\sqrt {x}}} \] With initial conditions \begin {align*} [y \left (\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {x}\, \ln \left (x \right ) c_{2} -\ln \left (x \right )+1}{\sqrt {x}} \]

Problem 15436

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

Problem 15437

ODE	\[ \boxed {x^{3} \left (\ln \left (x \right )-1\right ) y^{\prime \prime }-x^{2} y^{\prime }+y x=2 \ln \left (x \right )} \] With initial conditions \begin {align*} [y \left (\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \frac {-c_{1} \ln \left (x \right ) x +1}{x} \]

Problem 15438

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

Problem 15439

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

Problem 15440

ODE	\[ \boxed {x^{\prime \prime }+2 x^{\prime }+6 x=0} \]
program solution	\[ x = c_{1} {\mathrm e}^{-t} \cos \left (\sqrt {5}\, t \right )+\frac {c_{2} \sin \left (\sqrt {5}\, t \right ) {\mathrm e}^{-t} \sqrt {5}}{5} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-t} \left (c_{1} \sin \left (\sqrt {5}\, t \right )+c_{2} \cos \left (\sqrt {5}\, t \right )\right ) \]

Problem 15441

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

Problem 15442

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

Problem 15443

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

Problem 15444

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

Problem 15445

ODE	\[ \boxed {x^{\prime \prime }+{\mathrm e}^{-x^{\prime }}-x=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 15446

ODE	\[ \boxed {x^{\prime \prime }+x {x^{\prime }}^{2}=0} \]
program solution	\[ \int _{}^{x}\frac {{\mathrm e}^{\frac {\textit {\_a}^{2}}{2}}}{c_{1}}d \textit {\_a} = t +c_{2} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -i \operatorname {RootOf}\left (i \sqrt {2}\, c_{1} t +i \sqrt {2}\, c_{2} -\operatorname {erf}\left (\textit {\_Z} \right ) \sqrt {\pi }\right ) \sqrt {2} \]

Problem 15447

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

Problem 15448

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

Problem 15449

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

Problem 15450

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

Problem 15451

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

Problem 15452

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

Problem 15453

ODE	\[ \boxed {{y^{\prime }}^{2}+y y^{\prime \prime }=-1} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y \left (1\right ) = 2] \end {align*}
program solution	\[ \frac {y^{2}}{2}+\frac {x^{2}}{4}-\frac {7 x}{4}-\frac {1}{2} = 0 \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \sqrt {-x^{2}+4 x +1} \]

Problem 15454

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

Problem 15455

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

Problem 15456

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

Problem 15457

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

Problem 15458

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

Problem 15459

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

Problem 15460

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

Problem 15461

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

Problem 15462

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

Problem 15463

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

Problem 15464

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

Problem 15465

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

Problem 15466

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

Problem 15467

ODE	\[ \boxed {y^{\prime }+y x=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x -\frac {x^{3}}{3}+\frac {x^{5}}{15}+O\left (x^{6}\right ) \] Verified OK. \[ y = x -\frac {x^{3}}{3}+\frac {x^{5}}{15}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15468

ODE	\[ \boxed {y^{\prime }-\frac {y-x}{y+x}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}+x +1+\frac {4 x^{3}}{3}-\frac {5 x^{4}}{2}+\frac {16 x^{5}}{3}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+x -x^{2}+\frac {4}{3} x^{3}-\frac {5}{2} x^{4}+\frac {16}{3} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15469

ODE	\[ \boxed {y^{\prime }-y \sin \left (x \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{2}}{2}+\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1+\frac {x^{2}}{2}+\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {1}{2} x^{2}+\frac {1}{12} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15470

ODE	\[ \boxed {y^{\prime \prime }+y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x -\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK. \[ y = x -\frac {x^{4}}{12}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x -\frac {1}{12} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15471

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } \sin \left (x \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x +\frac {x^{3}}{6}+\frac {x^{5}}{60}+O\left (x^{6}\right ) \] Verified OK. \[ y = x +\frac {x^{3}}{6}+\frac {x^{5}}{60}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x +\frac {1}{6} x^{3}+\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15472

ODE	\[ \boxed {x y^{\prime \prime }+y \sin \left (x \right )=x} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 1, y^{\prime }\left (\pi \right ) = 0] \end {align*} With the expansion point for the power series method at \(x = \pi \).
program solution	\[ y = 1+\frac {\left (x -\pi \right )^{2}}{2}+\frac {\left (x -\pi \right )^{3}}{6 \pi }-\frac {\left (x -\pi \right )^{4}}{12 \pi ^{2}}+\frac {\left (x -\pi \right )^{5}}{60 \pi }+\frac {\left (x -\pi \right )^{5}}{20 \pi ^{3}}-\frac {\left (x -\pi \right )^{6}}{180 \pi ^{2}}-\frac {\left (x -\pi \right )^{6}}{30 \pi ^{4}}+O\left (\left (x -\pi \right )^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\frac {1}{2} \left (-\pi +x \right )^{2}+\frac {1}{6} \frac {1}{\pi } \left (-\pi +x \right )^{3}-\frac {1}{12} \frac {1}{\pi ^{2}} \left (-\pi +x \right )^{4}+\frac {1}{60} \frac {\pi ^{2}+3}{\pi ^{3}} \left (-\pi +x \right )^{5}+\operatorname {O}\left (\left (-\pi +x \right )^{6}\right ) \]

Problem 15473

ODE	\[ \boxed {\ln \left (x \right ) y^{\prime \prime }-y \sin \left (x \right )=0} \] With initial conditions \begin {align*} [y \left ({\mathrm e}\right ) = {\mathrm e}^{-1}, y^{\prime }\left ({\mathrm e}\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = {\mathrm e}\).
program solution	\[ y = {\mathrm e}^{-1}+\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{30}-\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2} \sin \left ({\mathrm e}\right )}{720}-\frac {11 \left (x -{\mathrm e}\right )^{6} \cos \left ({\mathrm e}\right ) \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{360}+O\left (\left (x -{\mathrm e}\right )^{6}\right )+\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{40}-\frac {7 \left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{60}-\frac {\left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{120}+\frac {3 \left (x -{\mathrm e}\right )^{5} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}+\frac {11 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \cos \left ({\mathrm e}\right )^{2}}{720}+\frac {5 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-3}}{144}+\frac {\left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{360}-\frac {7 \left (x -{\mathrm e}\right )^{6} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4}}{90}+\frac {\left (x -{\mathrm e}\right )^{6} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{180}-\frac {\left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{40}+\frac {11 \left (x -{\mathrm e}\right )^{6} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-5}}{90}+\frac {{\mathrm e}^{-1} \sin \left ({\mathrm e}\right ) \left (x -{\mathrm e}\right )^{2}}{2}-\frac {\left (x -{\mathrm e}\right )^{3} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{6}+\frac {\left (x -{\mathrm e}\right )^{3} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-1}}{6}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-1}}{24}-\frac {\left (x -{\mathrm e}\right )^{4} {\mathrm e}^{-1} \sin \left ({\mathrm e}\right )}{24}+\frac {\left (x -{\mathrm e}\right )^{4} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-3}}{8}-\frac {\left (x -{\mathrm e}\right )^{4} \cos \left ({\mathrm e}\right ) {\mathrm e}^{-2}}{12}-\frac {\left (x -{\mathrm e}\right )^{5} \sin \left ({\mathrm e}\right )^{2} {\mathrm e}^{-2}}{30}-\frac {7 \left (x -{\mathrm e}\right )^{6} {\mathrm e}^{-1}}{720} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-1}+\frac {1}{2} \sin \left ({\mathrm e}\right ) {\mathrm e}^{-1} \left (x -{\mathrm e}\right )^{2}+\frac {1}{6} \left (\cos \left ({\mathrm e}\right ) {\mathrm e}-\sin \left ({\mathrm e}\right )\right ) {\mathrm e}^{-2} \left (x -{\mathrm e}\right )^{3}+\left (\frac {{\mathrm e}^{-3} {\mathrm e}^{2} \sin \left ({\mathrm e}\right )^{2}}{24}-\frac {\left ({\mathrm e}^{2}-3\right ) {\mathrm e}^{-3} \sin \left ({\mathrm e}\right )}{24}-\frac {{\mathrm e}^{-3} \cos \left ({\mathrm e}\right ) {\mathrm e}}{12}\right ) \left (x -{\mathrm e}\right )^{4}+\left (-\frac {{\mathrm e}^{-4} {\mathrm e}^{2} \sin \left ({\mathrm e}\right )^{2}}{30}+\frac {\left (4 \cos \left ({\mathrm e}\right ) {\mathrm e}^{3}+3 \,{\mathrm e}^{2}-14\right ) {\mathrm e}^{-4} \sin \left ({\mathrm e}\right )}{120}+\frac {3 \cos \left ({\mathrm e}\right ) {\mathrm e}^{-4} \left ({\mathrm e}-\frac {{\mathrm e}^{3}}{9}\right )}{40}\right ) \left (x -{\mathrm e}\right )^{5}+\operatorname {O}\left (\left (x -{\mathrm e}\right )^{6}\right ) \]

Problem 15474

ODE
program solution
Maple solution	\[ y = \frac {\pi }{2}-\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15475

ODE	\[ \boxed {y^{\prime }-2 y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x^{2}+1+\frac {x^{4}}{2}+O\left (x^{6}\right ) \] Verified OK. \[ y = x^{2}+1+\frac {x^{4}}{2}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = 1+x^{2}+\frac {1}{2} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15476

ODE	\[ \boxed {y^{\prime \prime }+x y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}-\frac {1}{48} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) c_{1} +\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{8} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {1}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 15477

ODE	\[ \boxed {y^{\prime \prime }-x y^{\prime }+y=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{2}+\frac {x^{4}}{24}+\frac {x^{6}}{240}+O\left (x^{6}\right ) \] Verified OK. \[ y = \frac {x^{2}}{2}+\frac {x^{4}}{24}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = \frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15478

ODE	\[ \boxed {y^{\prime \prime }-\left (x^{2}+1\right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2, y^{\prime }\left (0\right ) = 2] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = -x^{2}+2 x -2+\frac {x^{3}}{3}-\frac {x^{4}}{4}+\frac {7 x^{5}}{60}-\frac {x^{6}}{24}+O\left (x^{6}\right ) \] Verified OK. \[ y = -2-x^{2}-\frac {x^{4}}{4}+2 x +\frac {x^{3}}{3}+\frac {7 x^{5}}{60}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = -2+2 x -x^{2}+\frac {1}{3} x^{3}-\frac {1}{4} x^{4}+\frac {7}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15479

ODE	\[ \boxed {y^{\prime \prime }-y x^{2}+y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = 1+\frac {x^{4}}{12}-\frac {x^{5}}{60}+\frac {x^{6}}{360}+O\left (x^{6}\right ) \] Verified OK. \[ y = 1+\frac {x^{4}}{12}-\frac {x^{5}}{60}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = 1+\frac {1}{12} x^{4}-\frac {1}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15480

ODE	\[ \boxed {y^{\prime \prime }-y \,{\mathrm e}^{x}=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}+\frac {13}{720} x^{6}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\frac {1}{72} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) c_{1} +\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = \left (1+\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{24} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 15481

ODE	\[ \boxed {y^{\prime }-{\mathrm e}^{y}-y x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*} With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x +\frac {x^{2}}{2}+\frac {2 x^{3}}{3}+\frac {11 x^{4}}{24}+\frac {53 x^{5}}{120}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = x +\frac {1}{2} x^{2}+\frac {2}{3} x^{3}+\frac {11}{24} x^{4}+\frac {53}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

Problem 15482

ODE	\[ \boxed {4 x y^{\prime \prime }+2 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {x}{6}+\frac {x^{2}}{120}-\frac {x^{3}}{5040}+\frac {x^{4}}{362880}-\frac {x^{5}}{39916800}+O\left (x^{6}\right )\right )+c_{2} \left (1-\frac {x}{2}+\frac {x^{2}}{24}-\frac {x^{3}}{720}+\frac {x^{4}}{40320}-\frac {x^{5}}{3628800}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y = c_{1} \sqrt {x}\, \left (1-\frac {1}{6} x +\frac {1}{120} x^{2}-\frac {1}{5040} x^{3}+\frac {1}{362880} x^{4}-\frac {1}{39916800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}-\frac {1}{720} x^{3}+\frac {1}{40320} x^{4}-\frac {1}{3628800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 15483

ODE	\[ \boxed {y^{\prime } \left (x +1\right )-n y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+n x +\frac {1}{2} n^{2} x^{2}-\frac {1}{2} n \,x^{2}+\frac {1}{6} n^{3} x^{3}-\frac {1}{2} n^{2} x^{3}+\frac {1}{3} n \,x^{3}+\frac {1}{24} n^{4} x^{4}-\frac {1}{4} n^{3} x^{4}+\frac {11}{24} n^{2} x^{4}-\frac {1}{4} n \,x^{4}+\frac {1}{120} n^{5} x^{5}-\frac {1}{12} n^{4} x^{5}+\frac {7}{24} n^{3} x^{5}-\frac {5}{12} n^{2} x^{5}+\frac {1}{5} n \,x^{5}\right ) y \left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+n x +\left (\frac {1}{2} n^{2}-\frac {1}{2} n \right ) x^{2}+\left (\frac {1}{6} n^{3}-\frac {1}{2} n^{2}+\frac {1}{3} n \right ) x^{3}+\left (\frac {1}{24} n^{4}-\frac {1}{4} n^{3}+\frac {11}{24} n^{2}-\frac {1}{4} n \right ) x^{4}+\left (\frac {1}{120} n^{5}-\frac {1}{12} n^{4}+\frac {7}{24} n^{3}-\frac {5}{12} n^{2}+\frac {1}{5} n \right ) x^{5}\right ) c_{1} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y = \left (1+n x +\frac {n \left (-1+n \right ) x^{2}}{2}+\frac {n \left (n^{2}-3 n +2\right ) x^{3}}{6}+\frac {n \left (n^{3}-6 n^{2}+11 n -6\right ) x^{4}}{24}+\frac {n \left (n^{4}-10 n^{3}+35 n^{2}-50 n +24\right ) x^{5}}{120}\right ) y \left (0\right )+O\left (x^{6}\right ) \]

Problem 15484

ODE	\[ \boxed {9 x \left (1-x \right ) y^{\prime \prime }-12 y^{\prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{\frac {7}{3}} \left (1+\frac {4 x}{5}+\frac {44 x^{2}}{65}+\frac {77 x^{3}}{130}+\frac {1309 x^{4}}{2470}+\frac {119 x^{5}}{247}+O\left (x^{6}\right )\right )+c_{2} \left (1+\frac {x}{3}+\frac {2 x^{2}}{9}+\frac {14 x^{3}}{81}+\frac {35 x^{4}}{243}+\frac {91 x^{5}}{729}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y = c_{1} x^{\frac {7}{3}} \left (1+\frac {4}{5} x +\frac {44}{65} x^{2}+\frac {77}{130} x^{3}+\frac {1309}{2470} x^{4}+\frac {119}{247} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+\frac {1}{3} x +\frac {2}{9} x^{2}+\frac {14}{81} x^{3}+\frac {35}{243} x^{4}+\frac {91}{729} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 15485

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

Problem 15486

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

Problem 15487

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {y}{9}=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (0, \frac {x}{3}\right )+c_{2} \operatorname {BesselY}\left (0, \frac {x}{3}\right ) \] Verified OK.
Maple solution	\[ y = c_{1} \operatorname {BesselJ}\left (0, \frac {x}{3}\right )+c_{2} \operatorname {BesselY}\left (0, \frac {x}{3}\right ) \]

Problem 15488

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

Problem 15489

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

Problem 15490

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

Problem 15491

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

Problem 15492

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

Problem 15493

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

Problem 15494

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

Problem 15495

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

Problem 15496

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

Problem 15497

ODE	\[ \boxed {y^{\prime \prime }+9 y=\sin \left (x \right )^{3}} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}+\frac {3 \sin \left (x \right )}{32}+\frac {x \cos \left (3 x \right )}{24} \] Verified OK.
Maple solution	\[ y = \frac {\left (x +24 c_{1} \right ) \cos \left (3 x \right )}{24}+\frac {\left (144 c_{2} -1\right ) \sin \left (3 x \right )}{144}+\frac {3 \sin \left (x \right )}{32} \]

Problem 15498

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&=-2 t x_{1} \left (t \right )^{2}\\ x_{2}^{\prime }\left (t \right )&=\frac {x_{2} \left (t \right )}{t}+1 \end {align*}
program solution
Maple solution	\begin{align*} \left \{x_{1} \left (t \right ) &= \frac {1}{t^{2}+c_{2}}\right \} \\ \{x_{2} \left (t \right ) &= \left (\ln \left (t \right )+c_{1} \right ) t\} \\ \end{align*}

Problem 15499

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&={\mathrm e}^{t} {\mathrm e}^{-x_{1} \left (t \right )}\\ x_{2}^{\prime }\left (t \right )&=2 \,{\mathrm e}^{x_{1} \left (t \right )} \end {align*}
program solution
Maple solution	\begin{align*} \{x_{1} \left (t \right ) &= \ln \left ({\mathrm e}^{t}+c_{2} \right )\} \\ \{x_{2} \left (t \right ) &= \int 2 \,{\mathrm e}^{x_{1} \left (t \right )}d t +c_{1}\} \\ \end{align*}

Problem 15500

ODE	\begin {align*} x^{\prime }\left (t \right )&=y \left (t \right )\\ y^{\prime }\left (t \right )&=\frac {y \left (t \right )^{2}}{x \left (t \right )} \end {align*}
program solution
Maple solution	\begin{align*} \{x \left (t \right ) &= {\mathrm e}^{c_{1} t} c_{2}\} \\ \{y \left (t \right ) &= \frac {d}{d t}x \left (t \right )\} \\ \end{align*}