2.17.28 Problems 2701 to 2800

Problem 2701

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

Problem 2702

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

Problem 2703

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

Problem 2704

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

Problem 2705

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

Problem 2706

ODE	\[ \boxed {y^{\prime }-\frac {x +2 y-1}{2 x -y+3}=0} \]
program solution	\[ -\frac {\ln \left (y^{2}+x^{2}-2 y+2 x +2\right )}{4}+\arctan \left (\frac {y-1}{x +1}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\tan \left (\operatorname {RootOf}\left (4 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x +1\right )+2 c_{1} \right )\right ) \left (-x -1\right ) \]

Problem 2707

ODE	\[ \boxed {y^{\prime }+p \left (x \right ) y+q \left (x \right ) y^{2}=r \left (x \right )} \]
program solution	\[ y = \frac {\frac {d}{d x}\operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )}{q \left (x \right ) \operatorname {DESol}\left (\left \{-q \left (x \right ) r \left (x \right ) \textit {\_Y} \left (x \right )+\frac {\left (p \left (x \right ) q \left (x \right )-q^{\prime }\left (x \right )\right ) \textit {\_Y}^{\prime }\left (x \right )}{q \left (x \right )}+\textit {\_Y}^{\prime \prime }\left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 2708

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

Problem 2709

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

Problem 2710

ODE	\[ \boxed {p \left (x \right ) \ln \left (y\right )=-\frac {y^{\prime }}{y}+q \left (x \right )} \]
program solution	\[ \int _{}^{x}\left (p \left (\textit {\_a} \right ) \ln \left (y\right )-q \left (\textit {\_a} \right )\right ) {\mathrm e}^{\int p \left (\textit {\_a} \right )d \textit {\_a}}d \textit {\_a} +\left (-{\mathrm e}^{\int _{}^{x}p \left (\textit {\_a} \right )d \textit {\_a}}+{\mathrm e}^{\int p \left (x \right )d x}\right ) \ln \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{{\mathrm e}^{-\left (\int p \left (x \right )d x \right )} \left (\int {\mathrm e}^{\int p \left (x \right )d x} q \left (x \right )d x -c_{1} \right )} \]

Problem 2711

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

Problem 2712

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

Problem 2713

ODE	\[ \boxed {y \,{\mathrm e}^{y x}+\left (2 y-x \,{\mathrm e}^{y x}\right ) y^{\prime }=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 2714

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

Problem 2715

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

Problem 2716

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

Problem 2717

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

Problem 2718

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

Problem 2719

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

Problem 2720

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

Problem 2721

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

Problem 2722

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

Problem 2723

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

Problem 2724

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

Problem 2725

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

Problem 2726

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

Problem 2727

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	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-6 x}+{\mathrm e}^{6 x} c_{2} \]

Problem 2728

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

Problem 2729

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

Problem 2730

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y^{\prime }-12 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{-3 x} c_{2} +c_{3} {\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{3} {\mathrm e}^{5 x}+{\mathrm e}^{x} c_{1} +c_{2} \right ) {\mathrm e}^{-3 x} \]

Problem 2731

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }-18 y^{\prime }-40 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{4 x} c_{2} +{\mathrm e}^{-5 x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{3} {\mathrm e}^{9 x}+c_{2} {\mathrm e}^{3 x}+c_{1} \right ) {\mathrm e}^{-5 x} \]

Problem 2732

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

Problem 2733

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }+8 y=0} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-4 x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{6 x} c_{2} +c_{1} {\mathrm e}^{5 x}+c_{3} \right ) {\mathrm e}^{-4 x} \]

Problem 2734

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

Problem 2735

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-13 y^{\prime \prime }+36 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+{\mathrm e}^{-3 x} c_{2} +c_{3} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{6 x}+c_{4} {\mathrm e}^{5 x}+c_{2} {\mathrm e}^{x}+c_{3} \right ) {\mathrm e}^{-3 x} \]

Problem 2736

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x -8 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{4}}+\frac {c_{2} x^{2}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{6}+c_{2}}{x^{4}} \]

Problem 2737

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

Problem 2738

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

Problem 2739

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

Problem 2740

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-6 y=18 \,{\mathrm e}^{5 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+\frac {c_{2} {\mathrm e}^{2 x}}{5}+\frac {3 \,{\mathrm e}^{5 x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (3 \,{\mathrm e}^{8 x}+4 c_{1} {\mathrm e}^{5 x}+4 c_{2} \right ) {\mathrm e}^{-3 x}}{4} \]

Problem 2741

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=4 x^{2}+5} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{x}}{3}-2 x^{2}-2 x -\frac {11}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-4 x^{2}-4 x -11\right ) {\mathrm e}^{-2 x} {\mathrm e}^{2 x}}{2}+\left (c_{1} {\mathrm e}^{3 x}+c_{2} \right ) {\mathrm e}^{-2 x} \]

Problem 2742

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

Problem 2743

ODE	\[ \boxed {y^{\prime \prime \prime }+y^{\prime \prime }-10 y^{\prime }+8 y=24 \,{\mathrm e}^{-3 x}} \]
program solution	\[ y = {\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-4 x} c_{3} +\frac {6 \,{\mathrm e}^{-3 x}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (5 c_{3} {\mathrm e}^{6 x}+5 c_{1} {\mathrm e}^{5 x}+6 \,{\mathrm e}^{x}+5 c_{2} \right ) {\mathrm e}^{-4 x}}{5} \]

Problem 2744

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

Problem 2745

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

Problem 2746

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

Problem 2747

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

Problem 2748

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

Problem 2749

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+5 y=3 \sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{-x} \sin \left (2 x \right )}{2}-\frac {12 \cos \left (2 x \right )}{17}+\frac {3 \sin \left (2 x \right )}{17} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (2 x \right ) \left (17 \,{\mathrm e}^{-x} c_{2} +3\right )}{17}+\cos \left (2 x \right ) {\mathrm e}^{-x} c_{1} -\frac {12 \cos \left (2 x \right )}{17} \]

Problem 2750

ODE	\[ \boxed {y^{\prime \prime \prime }+2 y^{\prime \prime }-5 y^{\prime }-6 y=4 x^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+{\mathrm e}^{-3 x} c_{2} +c_{3} {\mathrm e}^{2 x}-\frac {2 x^{2}}{3}+\frac {10 x}{9}-\frac {37}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-18 x^{2}+30 x -37\right ) {\mathrm e}^{-3 x} {\mathrm e}^{3 x}}{27}+\left (c_{2} {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{5 x}+c_{1} \right ) {\mathrm e}^{-3 x} \]

Problem 2751

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

Problem 2752

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

Problem 2753

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

Problem 2754

ODE	\[ \boxed {y^{\prime \prime }-y=9 x \,{\mathrm e}^{2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 7] \end {align*}
program solution	\[ y = \left (3 x -4\right ) {\mathrm e}^{2 x}+8 \,{\mathrm e}^{x}-4 \,{\mathrm e}^{-x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -4 \,{\mathrm e}^{-x}+8 \,{\mathrm e}^{x}+\left (3 x -4\right ) {\mathrm e}^{2 x} \]

Problem 2755

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

Problem 2756

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

Problem 2757

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

Problem 2758

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+6 y=7 \,{\mathrm e}^{2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x} \cos \left (\sqrt {2}\, x \right )+\frac {c_{2} {\mathrm e}^{2 x} \sqrt {2}\, \sin \left (\sqrt {2}\, x \right )}{2}+\frac {7 \,{\mathrm e}^{2 x}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (\frac {7}{2}+c_{2} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) c_{1} \right ) \]

Problem 2759

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

Problem 2760

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+104 y^{\prime \prime \prime }+2740 y^{\prime \prime }=5 \,{\mathrm e}^{-2 x} \cos \left (3 x \right )} \]
program solution	\[ y = c_{2} x +c_{1} +{\mathrm e}^{\left (-52-6 i\right ) x} c_{3} +{\mathrm e}^{\left (-52+6 i\right ) x} c_{4} -\frac {3475 \,{\mathrm e}^{-2 x} \cos \left (3 x \right )}{84184477}-\frac {12240 \,{\mathrm e}^{-2 x} \sin \left (3 x \right )}{84184477} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (667 c_{1} +156 c_{2} \right ) \cos \left (6 x \right )-156 \left (c_{1} -\frac {667 c_{2}}{156}\right ) \sin \left (6 x \right )\right ) {\mathrm e}^{-52 x}}{1876900}+\frac {5 \left (-695 \cos \left (3 x \right )-2448 \sin \left (3 x \right )\right ) {\mathrm e}^{-2 x}}{84184477}+c_{3} x +c_{4} \]

Problem 2761

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-3 y=\sin \left (x \right )^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+\frac {c_{2} {\mathrm e}^{x}}{4}-\frac {1}{6}+\frac {7 \cos \left (2 x \right )}{130}-\frac {2 \sin \left (2 x \right )}{65} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-3 x} \left (\left (-\frac {1}{6}-\frac {2 \sin \left (2 x \right )}{65}+\frac {7 \cos \left (2 x \right )}{130}\right ) {\mathrm e}^{3 x}+{\mathrm e}^{4 x} c_{1} +c_{2} \right ) \]

Problem 2762

ODE	\[ \boxed {y^{\prime \prime }+6 y=\sin \left (x \right )^{2} \cos \left (x \right )^{2}} \]
program solution	\[ y = c_{1} \cos \left (x \sqrt {6}\right )+\frac {c_{2} \sqrt {6}\, \sin \left (x \sqrt {6}\right )}{6}+\frac {1}{48}+\frac {\cos \left (4 x \right )}{80} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (\sqrt {6}\, x \right ) c_{2} +\cos \left (\sqrt {6}\, x \right ) c_{1} +\frac {1}{48}+\frac {\cos \left (4 x \right )}{80} \]

Problem 2763

ODE	\[ \boxed {y^{\prime \prime }-16 y=20 \cos \left (4 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 x}+\frac {{\mathrm e}^{4 x} c_{2}}{8}-\frac {5 \cos \left (4 x \right )}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{4 x}+{\mathrm e}^{-4 x} c_{1} -\frac {5 \cos \left (4 x \right )}{8} \]

Problem 2764

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+y=50 \sin \left (3 x \right )} \]
program solution	\[ y = {\mathrm e}^{-x} \left (c_{2} x +c_{1} \right )-3 \cos \left (3 x \right )-4 \sin \left (3 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-x}-3 \cos \left (3 x \right )-4 \sin \left (3 x \right ) \]

Problem 2765

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

Problem 2766

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=169 \sin \left (3 x \right )} \]
program solution	\[ y = {\mathrm e}^{-2 x} \left (c_{2} x +c_{1} \right )-12 \cos \left (3 x \right )-5 \sin \left (3 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{-2 x}-12 \cos \left (3 x \right )-5 \sin \left (3 x \right ) \]

Problem 2767

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }-2 y=40 \sin \left (x \right )^{2}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{2 x}}{3}-10+3 \cos \left (2 x \right )+\sin \left (2 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{2} +c_{1} {\mathrm e}^{2 x}-10+\sin \left (2 x \right )+3 \cos \left (2 x \right ) \]

Problem 2768

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

Problem 2769

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

Problem 2770

ODE	\[ \boxed {y^{\prime \prime }-4 y=100 \,{\mathrm e}^{x} x \sin \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{2 x}}{4}-14 \cos \left (x \right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} \sin \left (x \right )-10 \,{\mathrm e}^{x} x \cos \left (x \right )-20 \,{\mathrm e}^{x} x \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-10 x \cos \left (x \right ) {\mathrm e}^{3 x}+{\mathrm e}^{4 x} c_{1} -20 x \sin \left (x \right ) {\mathrm e}^{3 x}-14 \cos \left (x \right ) {\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x} \sin \left (x \right )+c_{2} \right ) {\mathrm e}^{-2 x} \]

Problem 2771

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

Problem 2772

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+10 y=24 \,{\mathrm e}^{x} \cos \left (3 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (3 x \right )+\frac {c_{2} {\mathrm e}^{x} \sin \left (3 x \right )}{3}+4 \sin \left (3 x \right ) x \,{\mathrm e}^{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{x} \left (3 c_{1} +4\right ) \cos \left (3 x \right )}{3}+4 \,{\mathrm e}^{x} \sin \left (3 x \right ) \left (x +\frac {c_{2}}{4}\right ) \]

Problem 2773

ODE	\[ \boxed {y^{\prime \prime }+16 y=34 \,{\mathrm e}^{x}+16 \cos \left (4 x \right )-8 \sin \left (4 x \right )} \]
program solution	\[ y = c_{1} \cos \left (4 x \right )+\frac {c_{2} \sin \left (4 x \right )}{4}+2 \,{\mathrm e}^{x}+x \cos \left (4 x \right )+2 x \sin \left (4 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (4 c_{2} +8 x -1\right ) \sin \left (4 x \right )}{4}+\left (c_{1} +x \right ) \cos \left (4 x \right )+2 \,{\mathrm e}^{x} \]

Problem 2774

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

Problem 2775

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

Problem 2776

ODE	\[ \boxed {y^{\prime \prime }+9 y=18 \sec \left (3 x \right )^{3}} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}-2 \cos \left (3 x \right )+\sec \left (3 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} -2\right ) \cos \left (3 x \right )+\sin \left (3 x \right ) c_{2} +\sec \left (3 x \right ) \]

Problem 2777

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

Problem 2778

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

Problem 2779

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

Problem 2780

ODE	\[ \boxed {y^{\prime \prime }+9 y=\frac {36}{4-\cos \left (3 x \right )^{2}}} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}+\left (\ln \left (\cos \left (3 x \right )+2\right )-\ln \left (\cos \left (3 x \right )-2\right )\right ) \cos \left (3 x \right )+\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \sin \left (3 x \right )}{3}\right ) \sin \left (3 x \right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \sin \left (3 x \right )}{3}\right ) \sin \left (3 x \right )}{3}+\sin \left (3 x \right ) c_{2} +\cos \left (3 x \right ) \ln \left (\cos \left (3 x \right )+2\right )-\cos \left (3 x \right ) \ln \left (\cos \left (3 x \right )-2\right )+\cos \left (3 x \right ) c_{1} \]

Problem 2781

ODE	\[ \boxed {y^{\prime \prime }-10 y^{\prime }+25 y=\frac {2 \,{\mathrm e}^{5 x}}{x^{2}+4}} \]
program solution	\[ y = {\mathrm e}^{5 x} \left (c_{2} x +c_{1} \right )+{\mathrm e}^{5 x} \left (-\ln \left (x^{2}+4\right )+x \arctan \left (\frac {x}{2}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{5 x} \left (c_{2} +c_{1} x -\ln \left (x^{2}+4\right )+x \arctan \left (\frac {x}{2}\right )\right ) \]

Problem 2782

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

Problem 2783

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

Problem 2784

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

Problem 2785

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

Problem 2786

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

Problem 2787

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

Problem 2788

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

Problem 2789

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+17 y=\frac {64 \,{\mathrm e}^{-x}}{3+\sin \left (4 x \right )^{2}}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x} \cos \left (4 x \right )+\frac {c_{2} {\mathrm e}^{-x} \sin \left (4 x \right )}{4}+\frac {4 \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \sin \left (4 x \right )}{3}\right ) \sin \left (4 x \right )-\frac {3 \cos \left (4 x \right ) \left (-\ln \left (\cos \left (4 x \right )+2\right )+\ln \left (\cos \left (4 x \right )-2\right )\right )}{4}\right ) {\mathrm e}^{-x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-x} \left (4 \sin \left (4 x \right ) \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \sin \left (4 x \right )}{3}\right )+3 \ln \left (\cos \left (4 x \right )+2\right ) \cos \left (4 x \right )-3 \ln \left (\cos \left (4 x \right )-2\right ) \cos \left (4 x \right )+3 \cos \left (4 x \right ) c_{1} +3 \sin \left (4 x \right ) c_{2} \right )}{3} \]

Problem 2790

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

Problem 2791

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+4 y=15 \ln \left (x \right ) {\mathrm e}^{-2 x}+25 \cos \left (x \right )} \]
program solution	\[ y = {\mathrm e}^{-2 x} \left (c_{2} x +c_{1} \right )+\frac {15 x^{2} \left (\ln \left (x \right )-\frac {3}{2}\right ) {\mathrm e}^{-2 x}}{2}+3 \cos \left (x \right )+4 \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (30 \ln \left (x \right ) x^{2}-45 x^{2}+4 c_{1} x +4 c_{2} \right ) {\mathrm e}^{-2 x}}{4}+3 \cos \left (x \right )+4 \sin \left (x \right ) \]

Problem 2792

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

Problem 2793

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y=36 \,{\mathrm e}^{2 x} \ln \left (x \right )} \]
program solution	\[ y = {\mathrm e}^{2 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right )+x^{3} {\mathrm e}^{2 x} \left (-11+6 \ln \left (x \right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (6 x^{3} \ln \left (x \right )-11 x^{3}+c_{1} +c_{2} x +c_{3} x^{2}\right ) \]

Problem 2794

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

Problem 2795

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+9 y^{\prime }=12 \,{\mathrm e}^{3 x}} \]
program solution	\[ y = \left (c_{3} x +c_{2} \right ) {\mathrm e}^{3 x}+c_{1} +2 \,{\mathrm e}^{3 x} x^{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (4+18 x^{2}+3 \left (-4+c_{1} \right ) x -c_{1} +3 c_{2} \right ) {\mathrm e}^{3 x}}{9}+c_{3} \]

Problem 2796

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

Problem 2797

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+4 y=F \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 x}+\frac {c_{2} {\mathrm e}^{-x}}{3}-\frac {\left (\int _{0}^{x}F \left (\alpha \right ) {\mathrm e}^{4 \alpha }d \alpha \right ) {\mathrm e}^{-4 x}}{3}+\frac {\left (\int _{0}^{x}F \left (\alpha \right ) {\mathrm e}^{\alpha }d \alpha \right ) {\mathrm e}^{-x}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} c_{2} +{\mathrm e}^{-4 x} c_{1} +\frac {\left (\int {\mathrm e}^{x} F \left (x \right )d x \right ) {\mathrm e}^{-x}}{3}-\frac {\left (\int F \left (x \right ) {\mathrm e}^{4 x}d x \right ) {\mathrm e}^{-4 x}}{3} \]

Problem 2798

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

Problem 2799

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }-12 y=F \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-6 x}+\frac {c_{2} {\mathrm e}^{2 x}}{8}-\frac {\left (\int _{0}^{x}F \left (\alpha \right ) {\mathrm e}^{6 \alpha }d \alpha \right ) {\mathrm e}^{-6 x}}{8}+\frac {\left (\int _{0}^{x}{\mathrm e}^{-2 \alpha } F \left (\alpha \right )d \alpha \right ) {\mathrm e}^{2 x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (-\left (\int F \left (x \right ) {\mathrm e}^{-2 x}d x \right ) {\mathrm e}^{8 x}-8 c_{1} {\mathrm e}^{8 x}+\int F \left (x \right ) {\mathrm e}^{6 x}d x -8 c_{2} \right ) {\mathrm e}^{-6 x}}{8} \]

Problem 2800

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=5 x \,{\mathrm e}^{2 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = {\mathrm e}^{2 x} \left (1-2 x +\frac {5}{6} x^{3}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x} \left (5 x^{3}-12 x +6\right )}{6} \]