2.17.49 Problems 4801 to 4900

Problem 4801

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

Problem 4802

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

Problem 4803

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

Problem 4804

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }-6 y^{\prime }=0} \]
program solution	\[ y = c_{1} +{\mathrm e}^{-3 x} c_{2} +c_{3} {\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} {\mathrm e}^{5 x}+c_{1} {\mathrm e}^{3 x}+c_{3} \right ) {\mathrm e}^{-3 x} \]

Problem 4805

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-9 y^{\prime }-5 y=0} \]
program solution	\[ y = {\mathrm e}^{\left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}+\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1+\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}{2}-\frac {2}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}-1-\frac {i \sqrt {3}\, \left (\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}-\frac {4}{\left (-3+i \sqrt {55}\right )^{\frac {1}{3}}}\right )}{2}\right ) x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\left (-1-2 \sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+2 \sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )\right ) x}+c_{2} {\mathrm e}^{-2 \left (\sqrt {3}\, \cos \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+\sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )+\frac {1}{2}\right ) x}+c_{3} {\mathrm e}^{\left (4 \sin \left (\frac {\arctan \left (\frac {\sqrt {55}}{3}\right )}{3}+\frac {\pi }{6}\right )-1\right ) x} \]

Problem 4806

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

Problem 4807

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }=10} \]
program solution	\[ y = -\frac {5 x}{2}-\frac {c_{1}}{4}-\frac {5}{8}+c_{2} {\mathrm e}^{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{4 x} c_{1}}{4}-\frac {5 x}{2}+c_{2} \]

Problem 4808

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

Problem 4809

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

Problem 4810

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=24 \,{\mathrm e}^{-3 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{3 x}}{4}+2 \,{\mathrm e}^{-3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{6 x} c_{1} +c_{2} {\mathrm e}^{2 x}+2\right ) {\mathrm e}^{-3 x} \]

Problem 4811

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

Problem 4812

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=12 \,{\mathrm e}^{-x}} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right )+3 \,{\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-3 x}+3 \,{\mathrm e}^{-x} \]

Problem 4813

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

Problem 4814

ODE	\[ \boxed {y^{\prime \prime }-16 y=40 \,{\mathrm e}^{4 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 x}+\frac {c_{2} {\mathrm e}^{4 x}}{8}+\frac {5 \,{\mathrm e}^{4 x} \left (-1+8 x \right )}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (5 x +c_{2} \right ) {\mathrm e}^{4 x}+{\mathrm e}^{-4 x} c_{1} \]

Problem 4815

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

Problem 4816

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

Problem 4817

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+10 y=100 \cos \left (4 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{-x} \sin \left (3 x \right )}{3}-6 \cos \left (4 x \right )+8 \sin \left (4 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \sin \left (3 x \right ) c_{2} +{\mathrm e}^{-x} \cos \left (3 x \right ) c_{1} +8 \sin \left (4 x \right )-6 \cos \left (4 x \right ) \]

Problem 4818

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+12 y=80 \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x} \cos \left (2 \sqrt {2}\, x \right )+\frac {c_{2} {\mathrm e}^{-2 x} \sqrt {2}\, \sin \left (2 \sqrt {2}\, x \right )}{4}-5 \cos \left (2 x \right )+5 \sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x} \sin \left (2 x \sqrt {2}\right ) c_{2} +{\mathrm e}^{-2 x} \cos \left (2 x \sqrt {2}\right ) c_{1} +5 \sin \left (2 x \right )-5 \cos \left (2 x \right ) \]

Problem 4819

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

Problem 4820

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+25 y=120 \sin \left (5 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 x} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{-4 x} \sin \left (3 x \right )}{3}-3 \cos \left (5 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-4 x} \sin \left (3 x \right ) c_{2} +{\mathrm e}^{-4 x} \cos \left (3 x \right ) c_{1} -3 \cos \left (5 x \right ) \]

Problem 4821

ODE	\[ \boxed {5 y^{\prime \prime }+12 y^{\prime }+20 y=120 \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {6 x}{5}} \cos \left (\frac {8 x}{5}\right )+\frac {5 c_{2} {\mathrm e}^{-\frac {6 x}{5}} \sin \left (\frac {8 x}{5}\right )}{8}-5 \cos \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {6 x}{5}} \sin \left (\frac {8 x}{5}\right ) c_{2} +{\mathrm e}^{-\frac {6 x}{5}} \cos \left (\frac {8 x}{5}\right ) c_{1} -5 \cos \left (2 x \right ) \]

Problem 4822

ODE	\[ \boxed {y^{\prime \prime }+9 y=30 \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}-5 x \cos \left (3 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-5 x +c_{1} \right ) \cos \left (3 x \right )+c_{2} \sin \left (3 x \right ) \]

Problem 4823

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

Problem 4824

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+17 y=60 \,{\mathrm e}^{-4 x} \sin \left (5 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x} \cos \left (4 x \right )+\frac {c_{2} {\mathrm e}^{-x} \sin \left (4 x \right )}{4}+2 \,{\mathrm e}^{-4 x} \cos \left (5 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \sin \left (4 x \right ) c_{2} +{\mathrm e}^{-x} \cos \left (4 x \right ) c_{1} +2 \,{\mathrm e}^{-4 x} \cos \left (5 x \right ) \]

Problem 4825

ODE	\[ \boxed {4 y^{\prime \prime }+4 y^{\prime }+5 y=40 \,{\mathrm e}^{-\frac {3 x}{2}} \sin \left (2 x \right )} \]
program solution	\[ y = {\mathrm e}^{-\frac {x}{2}} \left (c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )\right )+2 \,{\mathrm e}^{-\frac {3 x}{2}} \cos \left (2 x \right )-{\mathrm e}^{-\frac {3 x}{2}} \sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 4 \cos \left (x \right )^{2} {\mathrm e}^{-\frac {3 x}{2}}-2 \,{\mathrm e}^{-\frac {3 x}{2}} \cos \left (x \right ) \sin \left (x \right )+{\mathrm e}^{-\frac {x}{2}} \cos \left (x \right ) c_{1} +{\mathrm e}^{-\frac {x}{2}} \sin \left (x \right ) c_{2} -2 \,{\mathrm e}^{-\frac {3 x}{2}} \]

Problem 4826

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+8 y=30 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {5 x}{2}\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{-2 x} \sin \left (2 x \right )}{2}+4 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {5 x}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x} \sin \left (2 x \right ) c_{2} +{\mathrm e}^{-2 x} \cos \left (2 x \right ) c_{1} +4 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {5 x}{2}\right ) \]

Problem 4827

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

Problem 4828

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

Problem 4829

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

Problem 4830

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=12 x \,{\mathrm e}^{3 x}} \]
program solution	\[ y = {\mathrm e}^{3 x} \left (c_{2} x +c_{1} \right )+2 x^{3} {\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (2 x^{3}+c_{1} x +c_{2} \right ) \]

Problem 4831

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=16 x^{2} {\mathrm e}^{-x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{3 x}}{4}-\frac {x \,{\mathrm e}^{-x}}{2}-x^{2} {\mathrm e}^{-x}-\frac {4 x^{3} {\mathrm e}^{-x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-8 x^{3}-6 x^{2}+6 c_{2} -3 x \right ) {\mathrm e}^{-x}}{6}+c_{1} {\mathrm e}^{3 x} \]

Problem 4832

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

Problem 4833

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

Problem 4834

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

Problem 4835

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

Problem 4836

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

Problem 4837

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=4 \,{\mathrm e}^{x}+\left (1-x \right ) \left ({\mathrm e}^{2 x}-1\right )} \]
program solution	\[ y = {\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )+{\mathrm e}^{x}-3+x -\frac {{\mathrm e}^{2 x} x}{9}+\frac {5 \,{\mathrm e}^{2 x}}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -3+\left (c_{1} x +c_{2} \right ) {\mathrm e}^{-x}+\frac {\left (-3 x +5\right ) {\mathrm e}^{2 x}}{27}+x +{\mathrm e}^{x} \]

Problem 4838

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

Problem 4839

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

Problem 4840

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

Problem 4841

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

Problem 4842

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

Problem 4843

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

Problem 4844

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

Problem 4845

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

Problem 4846

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

Problem 4847

ODE	\[ \boxed {-\frac {y^{\prime \prime }}{\left (1+y^{\prime }\right )^{\frac {3}{2}}}=-k} \]
program solution	\[ y = -\frac {c_{1}^{2} k^{2}-4 c_{2}^{2} k^{2}-8 c_{2} k^{2} x -4 x^{2} k^{2}-16}{2 k^{2} \left (c_{1} -2 c_{2} -2 x \right )} \] Verified OK. \[ y = -\frac {c_{1}^{2} k^{2}-4 c_{3}^{2} k^{2}-8 c_{3} k^{2} x -4 x^{2} k^{2}-16}{2 k^{2} \left (c_{1} -2 c_{3} -2 x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -x -\frac {4}{k^{2} \left (x +c_{1} \right )}+c_{2} \]

Problem 4848

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

Problem 4849

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

Problem 4850

ODE	\[ \boxed {x^{2} y^{\prime \prime }+7 y^{\prime } x +9 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{3}}+\frac {c_{2} \ln \left (x \right )}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \ln \left (x \right )+c_{1}}{x^{3}} \]

Problem 4851

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +6 y=0} \]
program solution	\[ y = c_{1} x^{1-i \sqrt {5}}-\frac {i c_{2} \sqrt {5}\, x^{1+i \sqrt {5}}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \sin \left (\sqrt {5}\, \ln \left (x \right )\right )+c_{2} \cos \left (\sqrt {5}\, \ln \left (x \right )\right )\right ) \]

Problem 4852

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

Problem 4853

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

Problem 4854

ODE	\[ \boxed {x^{2} y^{\prime \prime }-5 y^{\prime } x +9 y=2 x^{3}} \]
program solution	\[ y = x^{3} \left (c_{1} +c_{2} \ln \left (x \right )\right )+\ln \left (x \right )^{2} x^{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3} \left (c_{2} +c_{1} \ln \left (x \right )+\ln \left (x \right )^{2}\right ) \]

Problem 4855

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

Problem 4856

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

Problem 4857

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

Problem 4858

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

Problem 4859

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

Problem 4860

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

Problem 4861

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

Problem 4862

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

Problem 4863

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

Problem 4864

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

Problem 4865

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

Problem 4866

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

Problem 4867

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

Problem 4868

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

Problem 4869

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

Problem 4870

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

Problem 4871

ODE	\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +y=x} \]
program solution	\[ y = x \left (c_{1} +c_{2} \ln \left (x \right )\right )+\frac {\ln \left (x \right )^{2} x}{2} \] 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 4872

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

Problem 4873

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

Problem 4874

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

Problem 4875

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

Problem 4876

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+5 y=26 \,{\mathrm e}^{3 x}} \]
program solution	\[ y = {\mathrm e}^{-2 x} \left (c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )\right )+{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{5 x}+c_{2} \sin \left (x \right )+\cos \left (x \right ) c_{1} \right ) {\mathrm e}^{-2 x} \]

Problem 4877

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

Problem 4878

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

Problem 4879

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

Problem 4880

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

Problem 4881

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

Problem 4882

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

Problem 4883

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

Problem 4884

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

Problem 4885

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+13 y^{\prime \prime }-18 y^{\prime }+36 y=0} \]
program solution	\[ y = {\mathrm e}^{-3 i x} c_{1} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} +{\mathrm e}^{3 i x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{x} \sin \left (\sqrt {3}\, x \right )+c_{2} {\mathrm e}^{x} \cos \left (\sqrt {3}\, x \right )+c_{3} \sin \left (3 x \right )+c_{4} \cos \left (3 x \right ) \]

Problem 4886

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

Problem 4887

ODE	\[ \boxed {x \left (y y^{\prime \prime }+{y^{\prime }}^{2}\right )-y y^{\prime }=0} \]
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 4888

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

Problem 4889

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

Problem 4890

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

Problem 4891

ODE	\[ \boxed {y y^{\prime \prime }+{y^{\prime }}^{2}=-4} \] With initial conditions \begin {align*} [y \left (1\right ) = 3, y^{\prime }\left (1\right ) = 0] \end {align*}
program solution	\[ \frac {y^{2}}{2}+x^{2}-2 x -\frac {7}{2} = 0 \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \sqrt {-4 x^{2}+8 x +5} \]

Problem 4892

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

Problem 4893

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

Problem 4894

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

Problem 4895

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

Problem 4896

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

Problem 4897

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

Problem 4898

ODE	\[ \boxed {y^{\prime \prime }+4 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-2 x^{2}+\frac {2}{3} x^{4}-\frac {4}{45} x^{6}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {2}{15} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-2 x^{2}+\frac {2}{3} x^{4}\right ) c_{1} +\left (x -\frac {2}{3} x^{3}+\frac {2}{15} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-2 x^{2}+\frac {2}{3} x^{4}\right ) y \left (0\right )+\left (x -\frac {2}{3} x^{3}+\frac {2}{15} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 4899

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

Problem 4900

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