2.17.52 Problems 5101 to 5200

Problem 5101

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

Problem 5102

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

Problem 5103

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

Problem 5104

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

Problem 5105

ODE	\[ \boxed {3 y+\left (7 y-3 x +3\right ) y^{\prime }=7 x -7} \]
program solution	\[ \frac {5 \ln \left (x -1+y\right )}{3}+\frac {2 \ln \left (-x +y+1\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 5106

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

Problem 5107

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

Problem 5108

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

Problem 5109

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

Problem 5110

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

Problem 5111

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

Problem 5112

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

Problem 5113

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

Problem 5114

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

Problem 5115

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

Problem 5116

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

Problem 5117

ODE	\[ \boxed {y^{\prime }-y \tan \left (x \right )=\cos \left (x \right )-2 x \sin \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{6}\right ) = 0\right ] \end {align*}
program solution	\[ y = \sec \left (x \right ) \cos \left (x \right )^{2} x -\frac {\sec \left (x \right ) \pi }{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \cos \left (x \right ) x -\frac {\pi \sec \left (x \right )}{8} \]

Problem 5118

ODE	\[ \boxed {y^{\prime }-\frac {2 y x +y^{2}}{x^{2}+2 y x}=0} \]
program solution	\[ -\frac {3 \left (x -y\right )}{\left (y x \right )^{\frac {1}{3}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {12^{\frac {1}{3}} \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}{6 c_{1}}+\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (\sqrt {3}\, \sqrt {\frac {x \left (27 c_{1} x -4\right )}{c_{1}}}+9 x \right ) c_{1}^{2}\right )^{\frac {1}{3}}}+x \\ y \left (x \right ) &= \frac {-\frac {\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {2}{3}}}{6}+\left (2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right )\right ) x c_{1}}{2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= -\frac {-\frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {2}{3}}}{6}+\left (-2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}}+2^{\frac {1}{3}} \left (i 3^{\frac {1}{6}}+\frac {3^{\frac {2}{3}}}{3}\right )\right ) x c_{1}}{2 {\left (x \left (\sqrt {3}\, \sqrt {\frac {27 c_{1} x^{2}-4 x}{c_{1}}}+9 x \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 5119

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

Problem 5120

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

Problem 5121

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

Problem 5122

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

Problem 5123

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

Problem 5124

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

Problem 5125

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

Problem 5126

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

Problem 5127

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

Problem 5128

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

Problem 5129

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

Problem 5130

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

Problem 5131

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

Problem 5132

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

Problem 5133

ODE	\[ \boxed {\frac {r \tan \left (\theta \right ) r^{\prime }}{a^{2}-r^{2}}=1} \] With initial conditions \begin {align*} \left [r \left (\frac {\pi }{4}\right ) = 0\right ] \end {align*}
program solution	\[ -\ln \left (\sin \left (\theta \right )\right )-\frac {\ln \left (r^{2}-a^{2}\right )}{2} = -\frac {\ln \left (-a^{2}\right )}{2}+\frac {\ln \left (2\right )}{2} \] Verified OK.
Maple solution	\begin{align*} r \left (\theta \right ) &= -\frac {a \sqrt {2}\, \sqrt {-\cos \left (2 \theta \right )}\, \csc \left (\theta \right )}{2} \\ r \left (\theta \right ) &= \frac {a \sqrt {2}\, \sqrt {-\cos \left (2 \theta \right )}\, \csc \left (\theta \right )}{2} \\ \end{align*}

Problem 5134

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

Problem 5135

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

Problem 5136

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

Problem 5137

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

Problem 5138

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

Problem 5139

ODE	\[ \boxed {y^{\prime \prime }+25 y=5 x^{2}+x} \]
program solution	\[ y = c_{1} \cos \left (5 x \right )+\frac {c_{2} \sin \left (5 x \right )}{5}+\frac {x^{2}}{5}+\frac {x}{25}-\frac {2}{125} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (5 x \right ) c_{2} +\cos \left (5 x \right ) c_{1} +\frac {x^{2}}{5}+\frac {x}{25}-\frac {2}{125} \]

Problem 5140

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

Problem 5141

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

Problem 5142

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

Problem 5143

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

Problem 5144

ODE	\[ \boxed {2 y^{\prime \prime }-7 y^{\prime }-4 y={\mathrm e}^{3 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}}+\frac {2 c_{2} {\mathrm e}^{4 x}}{9}-\frac {{\mathrm e}^{3 x}}{7} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} c_{2} +{\mathrm e}^{4 x} c_{1} -\frac {{\mathrm e}^{3 x}}{7} \]

Problem 5145

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=54 x +18} \]
program solution	\[ y = {\mathrm e}^{3 x} \left (c_{2} x +c_{1} \right )+6 x +6 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} x +c_{2} \right ) {\mathrm e}^{3 x}+6 x +6 \]

Problem 5146

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=100 \sin \left (4 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{3 x}+4 \cos \left (4 x \right )-2 \sin \left (4 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{3 x} c_{2} +{\mathrm e}^{2 x} c_{1} -2 \sin \left (4 x \right )+4 \cos \left (4 x \right ) \]

Problem 5147

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

Problem 5148

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-2 y=2 \cosh \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+\frac {c_{2} {\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{-2 x} \left (\left (\frac {4 \cosh \left (x \right )^{3}}{3}-\frac {4 \sinh \left (x \right ) \cosh \left (x \right )^{2}}{3}-2 \cosh \left (x \right )-\frac {2 \sinh \left (x \right )}{3}\right ) {\mathrm e}^{3 x}+2 \cosh \left (x \right )^{4}+2 \cosh \left (x \right )^{3} \sinh \left (x \right )-2 \cosh \left (x \right )^{2}-\cosh \left (x \right ) \sinh \left (x \right )+x +\frac {1}{2}\right )}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (9 \,{\mathrm e}^{4 x}+36 \,{\mathrm e}^{3 x} c_{2} +36 c_{1} -12 x -7\right ) {\mathrm e}^{-2 x}}{36} \]

Problem 5149

ODE	\[ \boxed {y^{\prime \prime }-y^{\prime }+10 y=20-{\mathrm e}^{2 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {39}\, x}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {39}\, x}{2}\right ) {\mathrm e}^{\frac {x}{2}} \sqrt {39}}{39}+2-\frac {{\mathrm e}^{2 x}}{12} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x}{2}} \sin \left (\frac {\sqrt {39}\, x}{2}\right ) c_{2} +{\mathrm e}^{\frac {x}{2}} \cos \left (\frac {\sqrt {39}\, x}{2}\right ) c_{1} +2-\frac {{\mathrm e}^{2 x}}{12} \]

Problem 5150

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

Problem 5151

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

Problem 5152

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+3 y=x^{2}-1} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (\sqrt {2}\, x \right )+\frac {c_{2} \sin \left (\sqrt {2}\, x \right ) \sqrt {2}\, {\mathrm e}^{x}}{2}+\frac {x^{2}}{3}+\frac {4 x}{9}-\frac {7}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \sin \left (\sqrt {2}\, x \right ) c_{2} +{\mathrm e}^{x} \cos \left (\sqrt {2}\, x \right ) c_{1} +\frac {x^{2}}{3}+\frac {4 x}{9}-\frac {7}{27} \]

Problem 5153

ODE	\[ \boxed {y^{\prime \prime }-9 y={\mathrm e}^{3 x}+\sin \left (x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+\frac {c_{2} {\mathrm e}^{3 x}}{6}+\frac {\left (6 x -1\right ) {\mathrm e}^{3 x}}{36}-\frac {\sin \left (x \right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-1+6 x +36 c_{2} \right ) {\mathrm e}^{3 x}}{36}+{\mathrm e}^{-3 x} c_{1} -\frac {\sin \left (x \right )}{10} \]

Problem 5154

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

Problem 5155

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

Problem 5156

ODE	\[ \boxed {x^{\prime \prime }-3 x^{\prime }+2 x=\sin \left (t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {{\mathrm e}^{2 t}}{5}-\frac {{\mathrm e}^{t}}{2}+\frac {3 \cos \left (t \right )}{10}+\frac {\sin \left (t \right )}{10} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {{\mathrm e}^{2 t}}{5}+\frac {3 \cos \left (t \right )}{10}+\frac {\sin \left (t \right )}{10}-\frac {{\mathrm e}^{t}}{2} \]

Problem 5157

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=3 \sin \left (x \right )} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {9}{10}}, y^{\prime }\left (0\right ) = -{\frac {7}{10}}\right ] \end {align*}
program solution	\[ y = -{\mathrm e}^{-x}+{\mathrm e}^{-2 x}-\frac {9 \cos \left (x \right )}{10}+\frac {3 \sin \left (x \right )}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-2 x}-\frac {9 \cos \left (x \right )}{10}+\frac {3 \sin \left (x \right )}{10}-{\mathrm e}^{-x} \]

Problem 5158

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+10 y=50 x} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )\right )+5 x -3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-3 x} \sin \left (x \right ) c_{2} +{\mathrm e}^{-3 x} \cos \left (x \right ) c_{1} +5 x -3 \]

Problem 5159

ODE	\[ \boxed {x^{\prime \prime }+2 x^{\prime }+2 x=85 \sin \left (3 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = -20] \end {align*}
program solution	\[ x = \left (6 \cos \left (t \right )+7 \sin \left (t \right )\right ) {\mathrm e}^{-t}-6 \cos \left (3 t \right )-7 \sin \left (3 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = \left (7 \sin \left (t \right )+6 \cos \left (t \right )\right ) {\mathrm e}^{-t}-6 \cos \left (3 t \right )-7 \sin \left (3 t \right ) \]

Problem 5160

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

Problem 5161

ODE	\[ \boxed {\frac {x^{\prime \prime }}{2}+48 x=0} \] With initial conditions \begin {align*} \left [x \left (0\right ) = {\frac {1}{6}}, x^{\prime }\left (0\right ) = 0\right ] \end {align*}
program solution	\[ x = \frac {\cos \left (4 \sqrt {6}\, t \right )}{6} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\cos \left (4 \sqrt {6}\, t \right )}{6} \]

Problem 5162

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

Problem 5163

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

Problem 5164

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

Problem 5165

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

Problem 5166

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+25 y=2 \sin \left (\frac {t}{2}\right )-\cos \left (\frac {t}{2}\right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t} \cos \left (4 t \right )+\frac {c_{2} {\mathrm e}^{3 t} \sin \left (4 t \right )}{4}-\frac {20 \cos \left (\frac {t}{2}\right )}{663}+\frac {56 \sin \left (\frac {t}{2}\right )}{663} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{3 t} \sin \left (4 t \right ) c_{2} +{\mathrm e}^{3 t} \cos \left (4 t \right ) c_{1} +\frac {56 \sin \left (\frac {t}{2}\right )}{663}-\frac {20 \cos \left (\frac {t}{2}\right )}{663} \]

Problem 5167

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+25 y=64 \,{\mathrm e}^{-t}} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t} \cos \left (4 t \right )+\frac {c_{2} {\mathrm e}^{3 t} \sin \left (4 t \right )}{4}+2 \,{\mathrm e}^{-t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{3 t} \sin \left (4 t \right ) c_{2} +{\mathrm e}^{3 t} \cos \left (4 t \right ) c_{1} +2 \,{\mathrm e}^{-t} \]

Problem 5168

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+25 y=50 t^{3}-36 t^{2}-63 t +18} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t} \cos \left (4 t \right )+\frac {c_{2} {\mathrm e}^{3 t} \sin \left (4 t \right )}{4}+2 t^{3}-3 t \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{3 t} \sin \left (4 t \right ) c_{2} +{\mathrm e}^{3 t} \cos \left (4 t \right ) c_{1} +2 t^{3}-3 t \]

Problem 5169

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y=2 x \,{\mathrm e}^{-x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{3 x} c_{3} -\frac {x \,{\mathrm e}^{-x}}{12}-\frac {13 \,{\mathrm e}^{-x}}{144} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-12 x -13\right ) {\mathrm e}^{-x}}{144}+{\mathrm e}^{x} c_{1} +c_{2} {\mathrm e}^{2 x}+c_{3} {\mathrm e}^{3 x} \]

Problem 5170

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

Problem 5171

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

Problem 5172

ODE	\[ \boxed {y^{\prime }-5 y=\left (x -1\right ) \sin \left (x \right )+\left (x +1\right ) \cos \left (x \right )} \]
program solution	\[ y = -\frac {{\mathrm e}^{5 x} \left (78 \,{\mathrm e}^{-5 x} x \cos \left (x \right )+52 \,{\mathrm e}^{-5 x} x \sin \left (x \right )+69 \,{\mathrm e}^{-5 x} \cos \left (x \right )-71 \,{\mathrm e}^{-5 x} \sin \left (x \right )-338 c_{1} \right )}{338} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{5 x}+\frac {\left (-78 x -69\right ) \cos \left (x \right )}{338}+\frac {\left (-52 x +71\right ) \sin \left (x \right )}{338} \]

Problem 5173

ODE	\[ \boxed {y^{\prime }-5 y=3 \,{\mathrm e}^{x}-2 x +1} \]
program solution	\[ y = -\frac {\left (-40 x \,{\mathrm e}^{-5 x}+75 \,{\mathrm e}^{-4 x}+12 \,{\mathrm e}^{-5 x}-100 c_{1} \right ) {\mathrm e}^{5 x}}{100} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 x}{5}-\frac {3}{25}-\frac {3 \,{\mathrm e}^{x}}{4}+c_{1} {\mathrm e}^{5 x} \]

Problem 5174

ODE	\[ \boxed {y^{\prime }-5 y={\mathrm e}^{x} x^{2}-x \,{\mathrm e}^{5 x}} \]
program solution	\[ y = -\frac {\left (8 x^{2} {\mathrm e}^{-4 x}+4 x \,{\mathrm e}^{-4 x}+16 x^{2}+{\mathrm e}^{-4 x}-32 c_{1} \right ) {\mathrm e}^{5 x}}{32} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (x^{2}-2 c_{1} \right ) {\mathrm e}^{x} {\mathrm e}^{4 x}}{2}+\frac {\left (-8 x^{2}-4 x -1\right ) {\mathrm e}^{x}}{32} \]

Problem 5175

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

Problem 5176

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

Problem 5177

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

Problem 5178

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

Problem 5179

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

Problem 5180

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

Problem 5181

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

Problem 5182

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

Problem 5183

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

Problem 5184

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

Problem 5185

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

Problem 5186

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

Problem 5187

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

Problem 5188

ODE	\[ \boxed {x^{\prime \prime }+4 x=\sin \left (2 t \right )^{2}} \]
program solution	\[ x = c_{1} \cos \left (2 t \right )+\frac {c_{2} \sin \left (2 t \right )}{2}+\frac {1}{8}+\frac {\cos \left (4 t \right )}{24} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \sin \left (2 t \right ) c_{2} +\cos \left (2 t \right ) c_{1} +\frac {1}{8}+\frac {\cos \left (4 t \right )}{24} \]

Problem 5189

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

Problem 5190

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

Problem 5191

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

Problem 5192

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

Problem 5193

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

Problem 5194

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

Problem 5195

ODE	\[ \boxed {y^{\prime \prime }-60 y^{\prime }-900 y=5 \,{\mathrm e}^{10 x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-30 \left (\sqrt {2}-1\right ) x}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{30 \left (1+\sqrt {2}\right ) x}}{120}-\frac {{\mathrm e}^{10 x}}{280} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{30 \left (1+\sqrt {2}\right ) x} c_{2} +{\mathrm e}^{-30 \left (\sqrt {2}-1\right ) x} c_{1} -\frac {{\mathrm e}^{10 x}}{280} \]

Problem 5196

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

Problem 5197

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

Problem 5198

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

Problem 5199

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

Problem 5200

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