2.17.47 Problems 4601 to 4700

Problem 4601

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

Problem 4602

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

Problem 4603

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

Problem 4604

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+20 y=0} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = 1, y^{\prime }\left (\frac {\pi }{2}\right ) = 1\right ] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{2 x -\pi } \left (4 \cos \left (4 x \right )-\sin \left (4 x \right )\right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\sin \left (4 x \right )-4 \cos \left (4 x \right )\right ) {\mathrm e}^{-\pi +2 x}}{4} \]

Problem 4605

ODE	\[ \boxed {3 y^{\prime \prime \prime }+5 y^{\prime \prime }+y^{\prime }-y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1, y^{\prime \prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {\left (9 \,{\mathrm e}^{\frac {4 x}{3}}+4 x -9\right ) {\mathrm e}^{-x}}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (9 \,{\mathrm e}^{\frac {4 x}{3}}+4 x -9\right ) {\mathrm e}^{-x}}{16} \]

Problem 4606

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

Problem 4607

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

Problem 4608

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

Problem 4609

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

Problem 4610

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

Problem 4611

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=8+6 \,{\mathrm e}^{x}+2 \sin \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{-x}+4+{\mathrm e}^{x}-\frac {3 \cos \left (x \right )}{5}+\frac {\sin \left (x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-2 x} \left (\left (-4+\frac {3 \cos \left (x \right )}{5}-\frac {\sin \left (x \right )}{5}\right ) {\mathrm e}^{2 x}-c_{2} {\mathrm e}^{x}+c_{1} -{\mathrm e}^{3 x}\right ) \]

Problem 4612

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

Problem 4613

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-8 y=9 x \,{\mathrm e}^{x}+10 \,{\mathrm e}^{-x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{4 x}}{6}-2 \,{\mathrm e}^{-x}-x \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{6 x} c_{1} -{\mathrm e}^{3 x} x -2 \,{\mathrm e}^{x}+c_{2} \right ) {\mathrm e}^{-2 x} \]

Problem 4614

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

Problem 4615

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

Problem 4616

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }=x +\sin \left (2 x \right )} \]
program solution	\[ y = \frac {x^{2}}{2}-\frac {2 \cos \left (x \right ) \sin \left (x \right )}{5}-\frac {\cos \left (x \right )^{2}}{5}+c_{1} -x +\frac {11}{10}+c_{2} {\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2}}{2}-{\mathrm e}^{-x} c_{1} -\frac {\sin \left (2 x \right )}{5}-\frac {\cos \left (2 x \right )}{10}-x +c_{2} \]

Problem 4617

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

Problem 4618

ODE	\[ \boxed {y^{\prime \prime }+4 y=x \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {x \sin \left (2 x \right )}{16}-\frac {\cos \left (2 x \right ) x^{2}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-x^{2}+8 c_{1} \right ) \cos \left (2 x \right )}{8}+\frac {\sin \left (2 x \right ) \left (16 c_{2} +x \right )}{16} \]

Problem 4619

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

Problem 4620

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

Problem 4621

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y=x \,{\mathrm e}^{-x}} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{2 x}+\frac {x \,{\mathrm e}^{-x}}{6}+\frac {5 \,{\mathrm e}^{-x}}{36} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (36 c_{1} {\mathrm e}^{3 x}+36 c_{2} {\mathrm e}^{2 x}+6 x +5\right ) {\mathrm e}^{-x}}{36} \]

Problem 4622

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=x +{\mathrm e}^{2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+\frac {c_{2} {\mathrm e}^{2 x}}{5}-\frac {1}{36}+\frac {\left (5 x -1\right ) {\mathrm e}^{2 x}}{25}-\frac {x}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (-\frac {6 x}{5}-6 c_{2} +\frac {6}{25}\right ) {\mathrm e}^{5 x}+\left (x +\frac {1}{6}\right ) {\mathrm e}^{3 x}-6 c_{1} \right ) {\mathrm e}^{-3 x}}{6} \]

Problem 4623

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

Problem 4624

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

Problem 4625

ODE	\[ \boxed {y^{\prime \prime }+y=\sin \left (2 x \right ) \sin \left (x \right )} \]
program solution	\[ y = \cos \left (x \right ) c_{1} +c_{2} \sin \left (x \right )+\frac {x \sin \left (x \right )}{4}+\frac {\cos \left (3 x \right )}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\sin \left (x \right )^{2} \cos \left (x \right )}{4}+\frac {\left (4 c_{2} +x \right ) \sin \left (x \right )}{4}+\cos \left (x \right ) c_{1} \]

Problem 4626

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }-6 y={\mathrm e}^{3 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {10 \,{\mathrm e}^{6 x}}{21}+\frac {45 \,{\mathrm e}^{-x}}{28}-\frac {{\mathrm e}^{3 x}}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {45 \,{\mathrm e}^{-x}}{28}+\frac {10 \,{\mathrm e}^{6 x}}{21}-\frac {{\mathrm e}^{3 x}}{12} \]

Problem 4627

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=5 \sin \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{2 x}}{3}+\frac {{\mathrm e}^{-x}}{6}+\frac {\cos \left (x \right )}{2}-\frac {3 \sin \left (x \right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x}}{6}+\frac {{\mathrm e}^{2 x}}{3}+\frac {\cos \left (x \right )}{2}-\frac {3 \sin \left (x \right )}{2} \]

Problem 4628

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

Problem 4629

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

Problem 4630

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

Problem 4631

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

Problem 4632

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

Problem 4633

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

Problem 4634

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

Problem 4635

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

Problem 4636

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

Problem 4637

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

Problem 4638

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

Problem 4639

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

Problem 4640

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

Problem 4641

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

Problem 4642

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

Problem 4643

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

Problem 4644

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

Problem 4645

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

Problem 4646

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

Problem 4647

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

Problem 4648

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

Problem 4649

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

Problem 4650

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

Problem 4651

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

Problem 4652

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

Problem 4653

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

Problem 4654

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

Problem 4655

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

Problem 4656

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

Problem 4657

ODE	\[ \boxed {r^{\prime \prime }+\frac {k}{r^{2}}=0} \]
program solution	\[ \frac {\sqrt {2}\, \left (\sqrt {r \left (c_{1} r+k \right )}\, \sqrt {c_{1}}-\operatorname {arctanh}\left (\frac {\sqrt {r \left (c_{1} r+k \right )}}{r \sqrt {c_{1}}}\right ) k \right )}{2 c_{1}^{\frac {3}{2}}} = t +c_{2} \] Verified OK. \[ -\frac {\sqrt {2}\, \left (\sqrt {r \left (c_{1} r+k \right )}\, \sqrt {c_{1}}-\operatorname {arctanh}\left (\frac {\sqrt {r \left (c_{1} r+k \right )}}{r \sqrt {c_{1}}}\right ) k \right )}{2 c_{1}^{\frac {3}{2}}} = t +c_{3} \] Verified OK.
Maple solution	\begin{align*} r \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ r \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ \end{align*}

Problem 4658

ODE	\[ \boxed {y^{\prime \prime }-\frac {3 k y^{2}}{2}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{3} k +2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {1}{\sqrt {\textit {\_a}^{3} k +2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 \operatorname {WeierstrassP}\left (x +c_{1} , 0, c_{2}\right )}{k} \]

Problem 4659

ODE	\[ \boxed {y^{\prime \prime }-2 k y^{3}=0} \]
program solution	\[ \int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{4} k +2 c_{1}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {1}{\sqrt {\textit {\_a}^{4} k +2 c_{1}}}d \textit {\_a} = x +c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} \operatorname {JacobiSN}\left (\left (\sqrt {-k}\, x +c_{1} \right ) c_{2} , i\right ) \]

Problem 4660

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

Problem 4661

ODE	\[ \boxed {r^{\prime \prime }-\frac {h^{2}}{r^{3}}+\frac {k}{r^{2}}=0} \]
program solution	\[ \frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}-\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} = t +c_{2} \] Verified OK. \[ -\frac {\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}}{2 c_{1}}+\frac {k \ln \left (\frac {\left (2 c_{1} r+k \right ) \sqrt {2}}{2 \sqrt {c_{1}}}+\sqrt {2 r^{2} c_{1} +2 r k -h^{2}}\right ) \sqrt {2}}{4 c_{1}^{\frac {3}{2}}} = t +c_{3} \] Verified OK.
Maple solution	\begin{align*} r \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+h^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}-2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} -2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ r \left (t \right ) &= \frac {c_{1} \left (c_{1}^{2} k^{2}-2 k c_{1} {\mathrm e}^{\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+{\mathrm e}^{2 \operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}+h^{2}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{4} k^{2}+2 \textit {\_Z} \,c_{1}^{3} k \,{\mathrm e}^{\textit {\_Z}}-\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{2 \textit {\_Z}} c_{1}^{2}+\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) c_{1}^{2} h^{2}+2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} c_{2} +2 \,\operatorname {csgn}\left (\frac {1}{c_{1}}\right ) {\mathrm e}^{\textit {\_Z}} t \right )}}{2} \\ \end{align*}

Problem 4662

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

Problem 4663

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

Problem 4664

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

Problem 4665

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

Problem 4666

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

Problem 4667

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime } {\mathrm e}^{y}=0} \] With initial conditions \begin {align*} [y \left (3\right ) = 0, y^{\prime }\left (3\right ) = 1] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = -\ln \left (-x +4\right ) \]

Problem 4668

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

Problem 4669

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

Problem 4670

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

Problem 4671

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

Problem 4672

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

Problem 4673

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

Problem 4674

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

Problem 4675

ODE	\[ \boxed {-a y^{3}+y^{\prime }=\frac {b}{x^{\frac {3}{2}}}} \]
program solution	\[ -\frac {\ln \left (x \right )}{2} = \int _{}^{y \sqrt {x}}-\frac {1}{2 \textit {\_a}^{3} a +\textit {\_a} +2 b}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 a \,\textit {\_a}^{3}+\textit {\_a} +2 b}d \textit {\_a} \right )\right )}{\sqrt {x}} \]

Problem 4676

ODE	\[ \boxed {a x y^{3}+b y^{2}+y^{\prime }=0} \]
program solution	\[ \frac {2 \ln \left (3 \left (y-\frac {b}{3 a x}\right ) a x -b \right ) \sqrt {b^{2}+4 a}-\ln \left (9 a^{2} \left (y-\frac {b}{3 a x}\right )^{2} x^{2}+\left (3 b \left (y-\frac {b}{3 a x}\right ) x -9\right ) a -2 b^{2}\right ) \sqrt {b^{2}+4 a}+2 b \,\operatorname {arctanh}\left (\frac {6 \left (y-\frac {b}{3 a x}\right ) a x +b}{3 \sqrt {b^{2}+4 a}}\right )}{2 \sqrt {b^{2}+4 a}} = \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{\operatorname {RootOf}\left (2 \sqrt {b^{2}+4 a}\, b \,\operatorname {arctanh}\left (\frac {2 a \,{\mathrm e}^{\textit {\_Z}}+b}{\sqrt {b^{2}+4 a}}\right )-\ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) b^{2}+2 c_{1} b^{2}+2 \textit {\_Z} \,b^{2}-4 \ln \left (x^{2} \left (a \,{\mathrm e}^{2 \textit {\_Z}}+b \,{\mathrm e}^{\textit {\_Z}}-1\right )\right ) a +8 c_{1} a +8 \textit {\_Z} a \right )}}{x} \]

Problem 4677

ODE	\[ \boxed {y^{\prime }-x^{a} y^{3}+3 y^{2}-x^{-a} y=x^{-2 a}-a \,x^{-a -1}} \]
program solution	\[ \frac {-32 \left (a -1\right )^{\frac {-2+a}{a -1}} {\mathrm e}^{\frac {i \pi +2 x^{1-a}}{a -1}} \left (\left (x -\frac {x^{-1+2 a}}{4}\right ) 2^{\frac {-3 a +5}{a -1}}+\frac {x^{-1+2 a} 4^{\frac {1}{a -1}}}{32}\right ) \left (y+x^{-a}\right )^{2} \operatorname {WhittakerM}\left (-\frac {1}{a -1}, \frac {a -3}{2 a -2}, -\frac {4 x^{1-a}}{a -1}\right )-\left (\left (4 \left (y+x^{-a}\right )^{2} x^{2}+2 x^{a +1} \left (y+x^{-a}\right )^{2}+a +1\right ) {\mathrm e}^{\frac {4 x^{1-a}}{a -1}}+2 c_{1} \left (y+x^{-a}\right )^{2} \left (a +1\right )\right ) \left (a -3\right )}{2 \left (a +1\right ) \left (a -3\right ) \left (y+x^{-a}\right )^{2}} = 0 \] Warning, solution could not be verified
Maple solution	\begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*}

Problem 4678

ODE	\[ \boxed {y^{\prime }-\left (y-f \left (x \right )\right ) \left (y-g \left (x \right )\right ) \left (y-\frac {a f \left (x \right )+b g \left (x \right )}{a +b}\right ) h \left (x \right )-\frac {f^{\prime }\left (x \right ) \left (y-g \left (x \right )\right )}{f \left (x \right )-g \left (x \right )}-\frac {g^{\prime }\left (x \right ) \left (y-f \left (x \right )\right )}{g \left (x \right )-f \left (x \right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\left (f \left (x \right )-g \left (x \right )\right ) \left (a +2 b \right ) {\mathrm e}^{\operatorname {RootOf}\left (-\textit {\_Z} \,a^{4}-b^{4} \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )+\ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{4}+\ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) b^{4}+3 c_{1} a^{3} b +6 c_{1} a^{2} b^{2}+3 c_{1} a \,b^{3}-2 a \,b^{3} \left (\int f \left (x \right ) g \left (x \right ) h \left (x \right )d x \right )-2 a^{3} b \left (\int f \left (x \right ) g \left (x \right ) h \left (x \right )d x \right )-2 a^{2} b^{2} \left (\int f \left (x \right ) g \left (x \right ) h \left (x \right )d x \right )+a^{2} b^{2} \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a \,b^{3} \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a^{3} b \left (\int f \left (x \right )^{2} h \left (x \right )d x \right )+a^{2} b^{2} \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )+a \,b^{3} \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )+a^{3} b \left (\int g \left (x \right )^{2} h \left (x \right )d x \right )-2 \textit {\_Z} \,a^{3} b -2 \textit {\_Z} \,a^{2} b^{2}-\textit {\_Z} a \,b^{3}+3 \ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{3} b +4 \ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a^{2} b^{2}+3 \ln \left (\frac {9 a^{2} b +9 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{a -b}\right ) a \,b^{3}-a^{3} b \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )-2 a^{2} b^{2} \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )-2 a \,b^{3} \ln \left (\frac {9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}+a \,{\mathrm e}^{\textit {\_Z}}+2 b \,{\mathrm e}^{\textit {\_Z}}}{2 a +b}\right )\right )}+9 f \left (x \right ) \left (a +b \right ) \left (a^{2}+b a +b^{2}\right )}{9 a^{3}+18 a^{2} b +18 a \,b^{2}+9 b^{3}} \]

Problem 4679

ODE	\[ \boxed {x^{2} y^{\prime }+x y^{3}+y^{2} a=0} \]
program solution
Maple solution	\[ c_{1} +{\mathrm e}^{-\frac {\left (\left (a +x \right ) y \left (x \right )+x \right ) \left (\left (a -x \right ) y \left (x \right )+x \right )}{2 y \left (x \right )^{2} x^{2}}} x +\frac {\operatorname {erf}\left (\frac {\sqrt {2}\, \left (a y \left (x \right )+x \right )}{2 y \left (x \right ) x}\right ) \sqrt {2}\, \sqrt {\pi }\, a \,{\mathrm e}^{\frac {1}{2}}}{2} = 0 \]

Problem 4680

ODE	\[ \boxed {\left (a x +b \right )^{2} y^{\prime }+\left (a x +b \right ) y^{3}+c y^{2}=0} \]
program solution
Maple solution	\[ \frac {\left (\sqrt {a}\, b +a^{\frac {3}{2}} x \right ) {\mathrm e}^{-\frac {\left (\left (a x +b +c \right ) y \left (x \right )+a \left (a x +b \right )\right ) \left (\left (-a x -b +c \right ) y \left (x \right )+a \left (a x +b \right )\right )}{2 y \left (x \right )^{2} \left (a x +b \right )^{2} a}}+\frac {c \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{\frac {1}{2 a}} \operatorname {erf}\left (\frac {\left (c y \left (x \right )+a \left (a x +b \right )\right ) \sqrt {2}}{2 \sqrt {a}\, y \left (x \right ) \left (a x +b \right )}\right )}{2}+c_{1} a^{\frac {3}{2}}}{a^{\frac {3}{2}}} = 0 \]

Problem 4681

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

Problem 4682

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

Problem 4683

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

Problem 4684

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

Problem 4685

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

Problem 4686

ODE	\[ \boxed {\left ({y^{\prime }}^{2}+1\right )^{3}-a^{2} {y^{\prime \prime }}^{2}=0} \]
program solution	\[ -\sqrt {\left (y+c_{1} +a \right ) \left (a -y-c_{1} \right )} = x +c_{3} \] Verified OK. \[ \sqrt {-y^{2}-2 y c_{1} -c_{1}^{2}+a^{2}} = x +c_{4} \] Verified OK. \[ -\sqrt {\left (a +y+c_{2} \right ) \left (a -y-c_{2} \right )} = x +c_{5} \] Verified OK. \[ \sqrt {-y^{2}-2 y c_{2} -c_{2}^{2}+a^{2}} = x +c_{6} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (-a +x +c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ y \left (x \right ) &= \frac {\left (a +x +c_{1} \right ) \left (a -x -c_{1} \right )}{\sqrt {a^{2}-c_{1}^{2}-2 c_{1} x -x^{2}}}+c_{2} \\ \end{align*}

Problem 4687

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

Problem 4688

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

Problem 4689

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

Problem 4690

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

Problem 4691

ODE	\[ \boxed {\frac {x}{1+y}-\frac {y y^{\prime }}{1+x}=0} \]
program solution	\[ -\frac {x^{3}}{3}+\frac {y^{3}}{3}-\frac {x^{2}}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {2}{3}}-i \sqrt {3}+2 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}+1}{4 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {2}{3}}-i \sqrt {3}-2 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}-1}{4 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 4692

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

Problem 4693

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

Problem 4694

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

Problem 4695

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

Problem 4696

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

Problem 4697

ODE	\[ \boxed {y^{\prime \prime }+y x=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{180} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{3}}{6}\right ) c_{1} +\left (x -\frac {1}{12} x^{4}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {x^{3}}{6}\right ) y \left (0\right )+\left (x -\frac {1}{12} x^{4}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 4698

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (-x^{2}+1\right ) y=x^{2}} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \frac {x^{2}}{3}+\frac {x^{4}}{63}+\frac {x^{6}}{3465}+O\left (x^{6}\right )+c_{1} x \left (1+\frac {x^{2}}{10}+\frac {x^{4}}{360}+O\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1+\frac {x^{2}}{6}+\frac {x^{4}}{168}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{63} x^{2}+\operatorname {O}\left (x^{4}\right )\right ) \]

Problem 4699

ODE	\[ \boxed {x y^{\prime \prime }+2 y^{\prime }+a^{3} x^{2} y=2} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = x -\frac {a^{3} x^{4}}{20}+O\left (x^{6}\right )+c_{1} \left (1-\frac {a^{3} x^{3}}{12}+O\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {a^{3} x^{3}}{6}+O\left (x^{6}\right )\right )}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (1-\frac {1}{12} a^{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-\frac {1}{6} a^{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x}+x \left (1-\frac {1}{20} a^{3} x^{3}+\operatorname {O}\left (x^{5}\right )\right ) \]

Problem 4700

ODE	\[ \boxed {y^{\prime \prime }+a \,x^{2} y=x +1} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {x^{4} a}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} a \,x^{5}\right ) y^{\prime }\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{6}-\frac {a \,x^{6}}{60}+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {x^{4} a}{12}\right ) c_{1} +\left (x -\frac {1}{20} a \,x^{5}\right ) c_{2} +\frac {x^{2}}{2}+\frac {x^{3}}{6}+O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {a \,x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} a \,x^{5}\right ) D\left (y \right )\left (0\right )+\frac {x^{2}}{2}+\frac {x^{3}}{6}+O\left (x^{6}\right ) \]