2.17.139 Problems 13801 to 13900

Problem 13801

ODE	\[ \boxed {y^{\prime \prime }-36 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-6 x}+\frac {c_{2} {\mathrm e}^{6 x}}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{6 x} c_{1} +{\mathrm e}^{-6 x} c_{2} \]

Problem 13802

ODE	\[ \boxed {y^{\prime \prime }-9 y^{\prime }+14 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{7 x}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{7 x}+c_{2} {\mathrm e}^{2 x} \]

Problem 13803

ODE	\[ \boxed {x^{2} y^{\prime \prime }-7 y^{\prime } x +16 y=0} \]
program solution	\[ y = c_{1} x^{4}+c_{2} x^{4} \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{4} \left (c_{1} +c_{2} \ln \left (x \right )\right ) \]

Problem 13804

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

Problem 13805

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

Problem 13806

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

Problem 13807

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

Problem 13808

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	\[ y \left (x \right ) = \frac {c_{1} +c_{2} \ln \left (x \right )}{x^{3}} \]

Problem 13809

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

Problem 13810

ODE	\[ \boxed {y^{\left (5\right )}-6 y^{\prime \prime \prime \prime }+13 y^{\prime \prime \prime }=0} \]
program solution	\[ y = x^{2} c_{3} +c_{2} x +c_{1} +{\mathrm e}^{\left (3+2 i\right ) x} c_{4} +{\mathrm e}^{\left (3-2 i\right ) x} c_{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +c_{2} x +x^{2} c_{3} +c_{4} {\mathrm e}^{3 x} \sin \left (2 x \right )+c_{5} {\mathrm e}^{3 x} \cos \left (2 x \right ) \]

Problem 13811

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

Problem 13812

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+25 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 x} \cos \left (4 x \right )+\frac {c_{2} {\mathrm e}^{3 x} \sin \left (4 x \right )}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} \left (c_{1} \sin \left (4 x \right )+c_{2} \cos \left (4 x \right )\right ) \]

Problem 13813

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

Problem 13814

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +9 y=0} \]
program solution	\[ y = x^{-3 i} c_{1} -\frac {i c_{2} x^{3 i}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (3 \ln \left (x \right )\right )+c_{2} \cos \left (3 \ln \left (x \right )\right ) \]

Problem 13815

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+25 y=0} \]
program solution	\[ y = c_{1} \cos \left (3 x \right ) {\mathrm e}^{4 x}+\frac {c_{2} \sin \left (3 x \right ) {\mathrm e}^{4 x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{4 x} \left (c_{1} \sin \left (3 x \right )+c_{2} \cos \left (3 x \right )\right ) \]

Problem 13816

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -30 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{6}}+\frac {c_{2} x^{5}}{11} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{5} c_{1} +\frac {c_{2}}{x^{6}} \]

Problem 13817

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-30 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-6 x}+\frac {c_{2} {\mathrm e}^{5 x}}{11} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{11 x} c_{1} +c_{2} \right ) {\mathrm e}^{-6 x} \]

Problem 13818

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

Problem 13819

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

Problem 13820

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }=8} \]
program solution	\[ y = c_{1} +{\mathrm e}^{\left (i \sqrt {3}+3\right ) x} c_{2} +{\mathrm e}^{\left (3-i \sqrt {3}\right ) x} c_{3} +\frac {2 x}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-\frac {\sqrt {3}\, c_{2}}{3}+c_{1} \right ) {\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )}{4}+\frac {{\mathrm e}^{3 x} \left (\sqrt {3}\, c_{1} +3 c_{2} \right ) \sin \left (\sqrt {3}\, x \right )}{12}+\frac {2 x}{3}+c_{3} \]

Problem 13821

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

Problem 13822

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

Problem 13823

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

Problem 13824

ODE	\[ \boxed {2 y^{\prime \prime }-7 y^{\prime }=-3} \]
program solution	\[ y = \frac {3 x}{7}-\frac {c_{1}}{7}+\frac {6}{49}+c_{2} {\mathrm e}^{\frac {7 x}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \,{\mathrm e}^{\frac {7 x}{2}} c_{1}}{7}+\frac {3 x}{7}+c_{2} \]

Problem 13825

ODE	\[ \boxed {y^{\prime \prime }+20 y^{\prime }+100 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-10 x}+c_{2} x \,{\mathrm e}^{-10 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-10 x} \left (c_{2} x +c_{1} \right ) \]

Problem 13826

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

Problem 13827

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

Problem 13828

ODE	\[ \boxed {y^{\prime \prime }-9 y^{\prime }+14 y=98 x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{7 x}}{5}+7 x^{2}+9 x +\frac {67}{14} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{7 x}+{\mathrm e}^{2 x} c_{1} +7 x^{2}+9 x +\frac {67}{14} \]

Problem 13829

ODE	\[ \boxed {y^{\prime \prime }-12 y^{\prime }+36 y=25 \sin \left (3 x \right )} \]
program solution	\[ y = {\mathrm e}^{6 x} \left (c_{2} x +c_{1} \right )+\frac {4 \cos \left (3 x \right )}{9}+\frac {\sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{6 x}+\frac {4 \cos \left (3 x \right )}{9}+\frac {\sin \left (3 x \right )}{3} \]

Problem 13830

ODE	\[ \boxed {y^{\prime \prime }-9 y^{\prime }+14 y=576 \,{\mathrm e}^{-x} x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+\frac {c_{2} {\mathrm e}^{7 x}}{5}+22 x \,{\mathrm e}^{-x}+24 \,{\mathrm e}^{-x} x^{2}+\frac {97 \,{\mathrm e}^{-x}}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{7 x}+{\mathrm e}^{2 x} c_{1} +\frac {\left (288 x^{2}+264 x +97\right ) {\mathrm e}^{-x}}{12} \]

Problem 13831

ODE	\[ \boxed {y^{\prime \prime }-12 y^{\prime }+36 y=81 \,{\mathrm e}^{3 x}} \]
program solution	\[ y = {\mathrm e}^{6 x} \left (c_{2} x +c_{1} \right )+9 \,{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{6 x}+9 \,{\mathrm e}^{3 x} \]

Problem 13832

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -9 y=3 \sqrt {x}} \]
program solution	\[ y = \frac {c_{1}}{x^{3}}+\frac {c_{2} x^{3}}{6}-\frac {12 \sqrt {x}}{35} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {35 c_{1} x^{6}-12 x^{\frac {7}{2}}+35 c_{2}}{35 x^{3}} \]

Problem 13833

ODE	\[ \boxed {y^{\prime \prime }-12 y^{\prime }+36 y=3 x \,{\mathrm e}^{6 x}-2 \,{\mathrm e}^{6 x}} \]
program solution	\[ y = {\mathrm e}^{6 x} \left (c_{2} x +c_{1} \right )-x^{2} {\mathrm e}^{6 x}+\frac {x^{3} {\mathrm e}^{6 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{6 x} \left (c_{2} +c_{1} x +\frac {1}{2} x^{3}-x^{2}+\frac {4}{9} x \right ) \]

Problem 13834

ODE	\[ \boxed {y^{\prime \prime }+36 y=6 \sec \left (6 x \right )} \]
program solution	\[ y = c_{1} \cos \left (6 x \right )+\frac {c_{2} \sin \left (6 x \right )}{6}-\frac {\ln \left (\sec \left (6 x \right )^{2}\right ) \cos \left (6 x \right )}{12}+x \sin \left (6 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\ln \left (\sec \left (6 x \right )\right ) \cos \left (6 x \right )}{6}+\cos \left (6 x \right ) c_{1} +\sin \left (6 x \right ) \left (c_{2} +x \right ) \]

Problem 13835

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -6 y=18 \ln \left (x \right )} \]
program solution	\[ y = \frac {c_{1}}{x^{3}}+\frac {c_{2} x^{2}}{5}-\frac {1}{2}-3 \ln \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2}}{x^{3}}+c_{1} x^{2}-3 \ln \left (x \right )-\frac {1}{2} \]

Problem 13836

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=10 \,{\mathrm e}^{-3 x}} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right )+5 x^{2} {\mathrm e}^{-3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-3 x} \left (c_{1} x +5 x^{2}+c_{2} \right ) \]

Problem 13837

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

Problem 13838

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=2 \cos \left (2 x \right )} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right )+\frac {10 \cos \left (2 x \right )}{169}+\frac {24 \sin \left (2 x \right )}{169} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-3 x}+\frac {10 \cos \left (2 x \right )}{169}+\frac {24 \sin \left (2 x \right )}{169} \]

Problem 13839

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

Problem 13840

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

Problem 13841

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

Problem 13842

ODE	\[ \boxed {4 y^{\prime \prime }-12 y^{\prime }+9 y=x \,{\mathrm e}^{\frac {3 x}{2}}} \]
program solution	\[ y = {\mathrm e}^{\frac {3 x}{2}} \left (c_{2} x +c_{1} \right )+\frac {x^{3} {\mathrm e}^{\frac {3 x}{2}}}{24} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {3 x}{2}} \left (c_{2} +c_{1} x +\frac {1}{24} x^{3}\right ) \]

Problem 13843

ODE	\[ \boxed {3 y^{\prime \prime }+8 y^{\prime }-3 y=123 \sin \left (3 x \right ) x} \]
program solution	\[ y = {\mathrm e}^{-3 x} c_{1} +\frac {3 c_{2} {\mathrm e}^{\frac {x}{3}}}{10}-2 \cos \left (3 x \right ) x -\frac {5 \sin \left (3 x \right ) x}{2}-\frac {241 \cos \left (3 x \right )}{246}+\frac {54 \sin \left (3 x \right )}{41} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2 \,{\mathrm e}^{-3 x} \left (-\frac {c_{1} {\mathrm e}^{\frac {10 x}{3}}}{2}+\left (\left (x +\frac {241}{492}\right ) \cos \left (3 x \right )+\sin \left (3 x \right ) \left (\frac {5 x}{4}-\frac {27}{41}\right )\right ) {\mathrm e}^{3 x}-\frac {c_{2}}{2}\right ) \]

Problem 13844

ODE	\[ \boxed {y^{\prime \prime \prime }+8 y={\mathrm e}^{-2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{2} +{\mathrm e}^{\left (1-i \sqrt {3}\right ) x} c_{3} +\frac {{\mathrm e}^{-2 x} x}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (12 c_{3} {\mathrm e}^{3 x} \sin \left (\sqrt {3}\, x \right )+12 c_{2} {\mathrm e}^{3 x} \cos \left (\sqrt {3}\, x \right )+12 c_{1} +x \right ) {\mathrm e}^{-2 x}}{12} \]

Problem 13845

ODE	\[ \boxed {y^{\left (6\right )}-64 y={\mathrm e}^{-2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{\left (1+i \sqrt {3}\right ) x} c_{3} +{\mathrm e}^{\left (i \sqrt {3}-1\right ) x} c_{4} +{\mathrm e}^{-\left (i \sqrt {3}-1\right ) x} c_{5} +{\mathrm e}^{-\left (1+i \sqrt {3}\right ) x} c_{6} -\frac {{\mathrm e}^{-2 x} x}{192} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (-192 c_{3} {\mathrm e}^{3 x}-192 c_{5} {\mathrm e}^{x}\right ) \cos \left (\sqrt {3}\, x \right )+\left (-192 c_{4} {\mathrm e}^{3 x}-192 c_{6} {\mathrm e}^{x}\right ) \sin \left (\sqrt {3}\, x \right )-192 c_{2} {\mathrm e}^{4 x}+x -192 c_{1} \right ) {\mathrm e}^{-2 x}}{192} \]

Problem 13846

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

Problem 13847

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

Problem 13848

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

Problem 13849

ODE	\[ \boxed {y^{\prime }-2 y=t^{3}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = -\frac {3 t^{2}}{4}-\frac {t^{3}}{2}-\frac {3 t}{4}+\frac {35 \,{\mathrm e}^{2 t}}{8}-\frac {3}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 t}{4}-\frac {t^{3}}{2}-\frac {3 t^{2}}{4}+\frac {35 \,{\mathrm e}^{2 t}}{8}-\frac {3}{8} \]

Problem 13850

ODE	\[ \boxed {y^{\prime }+3 y=\operatorname {Heaviside}\left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <4 \\ \frac {1}{3}-\frac {{\mathrm e}^{-3 t +12}}{3} & 4\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -4\right ) \left (-1+{\mathrm e}^{-3 t +12}\right )}{3} \]

Problem 13851

ODE	\[ \boxed {y^{\prime \prime }-4 y=t^{3}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {t^{3}}{4}+\frac {19 \,{\mathrm e}^{2 t}}{32}+\frac {13 \,{\mathrm e}^{-2 t}}{32}-\frac {3 t}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 t}{8}-\frac {t^{3}}{4}+\frac {19 \,{\mathrm e}^{2 t}}{32}+\frac {13 \,{\mathrm e}^{-2 t}}{32} \]

Problem 13852

ODE	\[ \boxed {y^{\prime \prime }+4 y=20 \,{\mathrm e}^{4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 12] \end {align*}
program solution	\[ y = {\mathrm e}^{4 t}+2 \cos \left (2 t \right )+4 \sin \left (2 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \cos \left (2 t \right )+4 \sin \left (2 t \right )+{\mathrm e}^{4 t} \]

Problem 13853

ODE	\[ \boxed {y^{\prime \prime }+4 y=\sin \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = \frac {21 \sin \left (2 t \right )}{8}-\frac {\cos \left (2 t \right ) \left (-12+t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {21 \sin \left (2 t \right )}{8}-\frac {\cos \left (2 t \right ) \left (-12+t \right )}{4} \]

Problem 13854

ODE	\[ \boxed {y^{\prime \prime }+4 y=3 \operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = \frac {5 \sin \left (2 t \right )}{2}+\left (\left \{\begin {array}{cc} 0 & t <2 \\ \frac {3 \sin \left (-2+t \right )^{2}}{2} & 2\le t \end {array}\right .\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {3 \operatorname {Heaviside}\left (t -2\right ) \sin \left (t -2\right )^{2}}{2}+\frac {5 \sin \left (2 t \right )}{2} \]

Problem 13855

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+6 y={\mathrm e}^{4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {\left ({\mathrm e}^{7 t}+119 \,{\mathrm e}^{t}-78\right ) {\mathrm e}^{-3 t}}{42} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left ({\mathrm e}^{7 t}+119 \,{\mathrm e}^{t}-78\right ) {\mathrm e}^{-3 t}}{42} \]

Problem 13856

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=t^{2} {\mathrm e}^{4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = -\frac {7 \,{\mathrm e}^{2 t}}{4}+\frac {{\mathrm e}^{4 t} \left (2 t^{2}-6 t +7\right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {7 \,{\mathrm e}^{2 t}}{4}+\frac {\left (2 t^{2}-6 t +7\right ) {\mathrm e}^{4 t}}{4} \]

Problem 13857

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=7} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = \frac {7}{6}-\frac {3 \,{\mathrm e}^{2 t}}{2}+\frac {7 \,{\mathrm e}^{3 t}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {7 \,{\mathrm e}^{3 t}}{3}-\frac {3 \,{\mathrm e}^{2 t}}{2}+\frac {7}{6} \]

Problem 13858

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+13 y=\sin \left (3 t \right ) {\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = -\frac {\left (-24+t \right ) {\mathrm e}^{2 t} \cos \left (3 t \right )}{6}-\frac {29 \sin \left (3 t \right ) {\mathrm e}^{2 t}}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (-24+t \right ) {\mathrm e}^{2 t} \cos \left (3 t \right )}{6}-\frac {29 \,{\mathrm e}^{2 t} \sin \left (3 t \right )}{18} \]

Problem 13859

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+13 y=4 t +2 \sin \left (3 t \right ) {\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = -\frac {16}{169}+\frac {2 \cosh \left (2 t \right ) \left (346 \cos \left (3 t \right )+313 \sin \left (3 t \right )\right )}{169}+\frac {\left (-1423 \cos \left (3 t \right )-1226 \sin \left (3 t \right )\right ) \sinh \left (2 t \right )}{338}+\frac {4 t}{13} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {16}{169}+\frac {2 \cosh \left (2 t \right ) \left (346 \cos \left (3 t \right )+313 \sin \left (3 t \right )\right )}{169}+\frac {\left (-1423 \cos \left (3 t \right )-1226 \sin \left (3 t \right )\right ) \sinh \left (2 t \right )}{338}+\frac {4 t}{13} \]

Problem 13860

ODE	\[ \boxed {y^{\prime \prime \prime }-27 y={\mathrm e}^{-3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 3, y^{\prime \prime }\left (0\right ) = 4] \end {align*}
program solution	\[ y = \frac {92 \cosh \left (3 t \right )}{81}+\frac {95 \sinh \left (3 t \right )}{81}+\frac {14 \,{\mathrm e}^{-\frac {3 t}{2}} \left (\sqrt {3}\, \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )+5 \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )\right )}{81} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {14 \sqrt {3}\, {\mathrm e}^{-\frac {3 t}{2}} \sin \left (\frac {3 \sqrt {3}\, t}{2}\right )}{81}+\frac {70 \,{\mathrm e}^{-\frac {3 t}{2}} \cos \left (\frac {3 \sqrt {3}\, t}{2}\right )}{81}+\frac {92 \cosh \left (3 t \right )}{81}+\frac {95 \sinh \left (3 t \right )}{81} \]

Problem 13861

ODE
program solution
Maple solution	\[ y \left (t \right ) = \operatorname {BesselJ}\left (0, t\right ) \]

Problem 13862

ODE	\[ \boxed {y^{\prime \prime }-9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 9] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-3 t}}{2}+\frac {7 \,{\mathrm e}^{3 t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-3 t}}{2}+\frac {7 \,{\mathrm e}^{3 t}}{2} \]

Problem 13863

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

Problem 13864

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+7 y=165 \,{\mathrm e}^{4 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 8, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \left (3 \,{\mathrm e}^{11 t}+4 \,{\mathrm e}^{6 t}+1\right ) {\mathrm e}^{-7 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (3 \,{\mathrm e}^{11 t}+4 \,{\mathrm e}^{6 t}+1\right ) {\mathrm e}^{-7 t} \]

Problem 13865

ODE	\[ \boxed {y^{\prime \prime }-8 y^{\prime }+17 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 12] \end {align*}
program solution	\[ y = 3 \cos \left (t \right ) {\mathrm e}^{4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = 3 \,{\mathrm e}^{4 t} \cos \left (t \right ) \]

Problem 13866

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y={\mathrm e}^{3 t} t^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {t^{4} {\mathrm e}^{3 t}}{12} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t^{4} {\mathrm e}^{3 t}}{12} \]

Problem 13867

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+13 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 8] \end {align*}
program solution	\[ y = {\mathrm e}^{-3 t} \left (2 \cos \left (2 t \right )+7 \sin \left (2 t \right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-3 t} \left (2 \cos \left (2 t \right )+7 \sin \left (2 t \right )\right ) \]

Problem 13868

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+17 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -12] \end {align*}
program solution	\[ y = 3 \,{\mathrm e}^{-4 t} \cos \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = 3 \,{\mathrm e}^{-4 t} \cos \left (t \right ) \]

Problem 13869

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

Problem 13870

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+40 y=122 \,{\mathrm e}^{-3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 8] \end {align*}
program solution	\[ y = -2 \,{\mathrm e}^{-3 t} \left (-1+\left (\cos \left (6 t \right )-\frac {3 \sin \left (6 t \right )}{2}\right ) {\mathrm e}^{5 t}\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -2 \left (-1+\left (\cos \left (6 t \right )-\frac {3 \sin \left (6 t \right )}{2}\right ) {\mathrm e}^{5 t}\right ) {\mathrm e}^{-3 t} \]

Problem 13871

ODE	\[ \boxed {y^{\prime \prime }-9 y=24 \,{\mathrm e}^{-3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 6, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \left (-4 t +2\right ) {\mathrm e}^{-3 t}+4 \,{\mathrm e}^{3 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-4 t +2\right ) {\mathrm e}^{-3 t}+4 \,{\mathrm e}^{3 t} \]

Problem 13872

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+13 y=\sin \left (3 t \right ) {\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 3] \end {align*}
program solution	\[ y = -\frac {\left (-24+t \right ) {\mathrm e}^{2 t} \cos \left (3 t \right )}{6}-\frac {29 \sin \left (3 t \right ) {\mathrm e}^{2 t}}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (-24+t \right ) {\mathrm e}^{2 t} \cos \left (3 t \right )}{6}-\frac {29 \,{\mathrm e}^{2 t} \sin \left (3 t \right )}{18} \]

Problem 13873

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

Problem 13874

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

Problem 13875

ODE	\[ \boxed {y^{\prime \prime }+4 y={\mathrm e}^{3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{3 t}}{13}-\frac {\cos \left (2 t \right )}{13}-\frac {3 \sin \left (2 t \right )}{26} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\cos \left (2 t \right )}{13}-\frac {3 \sin \left (2 t \right )}{26}+\frac {{\mathrm e}^{3 t}}{13} \]

Problem 13876

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

Problem 13877

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

Problem 13878

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

Problem 13879

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

Problem 13880

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

Problem 13881

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y={\mathrm e}^{-3 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {t \cosh \left (3 t \right )}{6}+\frac {\sinh \left (3 t \right ) \left (3 t -1\right )}{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t \cosh \left (3 t \right )}{6}+\frac {\sinh \left (3 t \right ) \left (3 t -1\right )}{18} \]

Problem 13882

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y={\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{t}}{4}+\frac {{\mathrm e}^{3 t} \left (2 t -1\right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{t}}{4}+\frac {\left (2 t -1\right ) {\mathrm e}^{3 t}}{4} \]

Problem 13883

ODE	\[ \boxed {y^{\prime }=\operatorname {Heaviside}\left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <3 \\ t -3 & 3\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -3\right ) \left (t -3\right ) \]

Problem 13884

ODE	\[ \boxed {y^{\prime }=\operatorname {Heaviside}\left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 4 & t <3 \\ 1+t & 3\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -3\right ) \left (t -3\right )+4 \]

Problem 13885

ODE	\[ \boxed {y^{\prime \prime }=\operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <2 \\ \frac {\left (-2+t \right )^{2}}{2} & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -2\right ) \left (t -2\right )^{2}}{2} \]

Problem 13886

ODE	\[ \boxed {y^{\prime \prime }=\operatorname {Heaviside}\left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 4, y^{\prime }\left (0\right ) = 6] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 4+6 t & t <2 \\ 6+4 t +\frac {1}{2} t^{2} & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = 4+\frac {\operatorname {Heaviside}\left (t -2\right ) \left (t -2\right )^{2}}{2}+6 t \]

Problem 13887

ODE	\[ \boxed {y^{\prime \prime }+9 y=\operatorname {Heaviside}\left (t -10\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <10 \\ \frac {1}{9}-\frac {\cos \left (3 t -30\right )}{9} & 10\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2 \operatorname {Heaviside}\left (t -10\right ) \sin \left (\frac {3 t}{2}-15\right )^{2}}{9} \]

Problem 13888

ODE	\[ \boxed {y^{\prime }=\left \{\begin {array}{cc} 0 & t <1 \\ 1 & 1
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ t -1 & t <3 \\ 2 & 3\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left \{\begin {array}{cc} 0 & t <1 \\ t -1 & t <3 \\ 2 & 3\le t \end {array}\right . \]

Problem 13889

ODE	\[ \boxed {y^{\prime \prime }=\left \{\begin {array}{cc} 0 & t <1 \\ 1 & 1
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {\left (t -1\right )^{2}}{2} & t <3 \\ 2 t -4 & 3\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left \{\begin {array}{cc} 0 & t <1 \\ \frac {\left (t -1\right )^{2}}{2} & t <3 \\ 2 t -4 & 3\le t \end {array}\right . \]

Problem 13890

ODE	\[ \boxed {y^{\prime \prime }+9 y=\left \{\begin {array}{cc} 0 & t <1 \\ 1 & 1
program solution	\[ y = -\frac {\left (\left \{\begin {array}{cc} 0 & t <1 \\ -1+\cos \left (3 t -3\right ) & t <3 \\ \cos \left (3 t -3\right )-\cos \left (3 t -9\right ) & 3\le t \end {array}\right .\right )}{9} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (\left \{\begin {array}{cc} 0 & t <1 \\ 1-\cos \left (3 t -3\right ) & t <3 \\ \cos \left (3 t -9\right )-\cos \left (3 t -3\right ) & 3\le t \end {array}\right .\right )}{9} \]

Problem 13891

ODE	\[ \boxed {y^{\prime }=3 \delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <2 \\ 3 & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = 3 \operatorname {Heaviside}\left (t -2\right ) \]

Problem 13892

ODE	\[ \boxed {y^{\prime }=\delta \left (-2+t \right )-\delta \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <2 \\ 1 & t <4 \\ 0 & 4\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -2\right ) \]

Problem 13893

ODE	\[ \boxed {y^{\prime \prime }=\delta \left (t -3\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <3 \\ t -3 & 3\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -3\right ) \left (t -3\right ) \]

Problem 13894

ODE	\[ \boxed {y^{\prime \prime }=\delta \left (t -1\right )-\delta \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ t -1 & t <4 \\ 3 & 4\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (4-t \right ) \operatorname {Heaviside}\left (t -4\right )+\operatorname {Heaviside}\left (t -1\right ) \left (t -1\right ) \]

Problem 13895

ODE	\[ \boxed {y^{\prime }+2 y=4 \delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <1 \\ 4 \,{\mathrm e}^{-2 t +2} & 1\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = 4 \operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-2 t +2} \]

Problem 13896

ODE	\[ \boxed {y^{\prime \prime }+y=\delta \left (t \right )+\delta \left (t -\pi \right )} \]
program solution	\[ y = c_{1} \cos \left (t \right )+\sin \left (t \right ) \left (c_{2} +\left (\left \{\begin {array}{cc} 1 & t \le \pi \\ 0 & \pi
Maple solution	\[ y \left (t \right ) = \cos \left (t \right ) y \left (0\right )+\sin \left (t \right ) \left (\operatorname {Heaviside}\left (\pi -t \right )+D\left (y \right )\left (0\right )\right ) \]

Problem 13897

ODE	\[ \boxed {y^{\prime \prime }+y=-2 \delta \left (t -\frac {\pi }{2}\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 0 & t <\frac {\pi }{2} \\ 2 \cos \left (t \right ) & \frac {\pi }{2}\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = 2 \cos \left (t \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \]

Problem 13898

ODE	\[ \boxed {y^{\prime }+3 y=\delta \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ y = \left \{\begin {array}{cc} 2 \,{\mathrm e}^{-3 t} & t <2 \\ 2 \,{\mathrm e}^{-3 t}+{\mathrm e}^{6-3 t} & 2\le t \end {array}\right . \] Verified OK.
Maple solution	\[ y \left (t \right ) = \operatorname {Heaviside}\left (t -2\right ) {\mathrm e}^{6-3 t}+2 \,{\mathrm e}^{-3 t} \]

Problem 13899

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

Problem 13900

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }=\delta \left (t -1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = -\frac {{\mathrm e}^{-3 t}}{3}+\frac {\left (\left \{\begin {array}{cc} 1 & t <1 \\ 2-{\mathrm e}^{-3 t +3} & 1\le t \end {array}\right .\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\operatorname {Heaviside}\left (t -1\right ) {\mathrm e}^{-3 t +3}}{3}+\frac {\operatorname {Heaviside}\left (t -1\right )}{3}-\frac {{\mathrm e}^{-3 t}}{3}+\frac {1}{3} \]