2.17.122 Problems 12101 to 12200

Problem 12101

ODE	\begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} -\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right )}{2} \\ \end{align*}

Problem 12102

ODE	\begin {align*} x^{\prime }\left (t \right )&=-2 x \left (t \right )+3 y \left (t \right )\\ y^{\prime }\left (t \right )&=-6 x \left (t \right )+4 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \sin \left (3 t \right )+c_{2} \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{t} \left (c_{1} \cos \left (3 t \right )+c_{2} \cos \left (3 t \right )+c_{1} \sin \left (3 t \right )-c_{2} \sin \left (3 t \right )\right ) \\ \end{align*}

Problem 12103

ODE	\begin {align*} x^{\prime }\left (t \right )&=-11 x \left (t \right )-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=13 x \left (t \right )-9 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-10 t} \left (\sin \left (5 t \right ) c_{1} +c_{2} \cos \left (5 t \right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-10 t} \left (\sin \left (5 t \right ) c_{1} -5 c_{2} \sin \left (5 t \right )+5 \cos \left (5 t \right ) c_{1} +c_{2} \cos \left (5 t \right )\right )}{2} \\ \end{align*}

Problem 12104

ODE	\begin {align*} x^{\prime }\left (t \right )&=7 x \left (t \right )-5 y \left (t \right )\\ y^{\prime }\left (t \right )&=10 x \left (t \right )-3 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (5 t \right ) c_{1} +c_{2} \cos \left (5 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \left (\sin \left (5 t \right ) c_{1} +c_{2} \sin \left (5 t \right )-\cos \left (5 t \right ) c_{1} +c_{2} \cos \left (5 t \right )\right ) \\ \end{align*}

Problem 12105

ODE	\begin {align*} x^{\prime }\left (t \right )&=5 x \left (t \right )-4 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{3 t} \left (2 c_{2} t +2 c_{1} -c_{2} \right )}{4} \\ \end{align*}

Problem 12106

ODE	\begin {align*} x^{\prime }\left (t \right )&=-6 x \left (t \right )+2 y \left (t \right )\\ y^{\prime }\left (t \right )&=-2 x \left (t \right )-2 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-4 t} \left (2 c_{2} t +2 c_{1} +c_{2} \right )}{2} \\ \end{align*}

Problem 12107

ODE	\begin {align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )-y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )-5 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= {\mathrm e}^{-4 t} \left (c_{2} t +c_{1} -c_{2} \right ) \\ \end{align*}

Problem 12108

ODE	\begin {align*} x^{\prime }\left (t \right )&=13 x \left (t \right )\\ y^{\prime }\left (t \right )&=13 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{13 t} \\ y \left (t \right ) &= {\mathrm e}^{13 t} c_{1} \\ \end{align*}

Problem 12109

ODE	\begin {align*} x^{\prime }\left (t \right )&=7 x \left (t \right )-4 y \left (t \right )\\ y^{\prime }\left (t \right )&=x \left (t \right )+3 y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{5 t} \left (2 c_{2} t +2 c_{1} -c_{2} \right )}{4} \\ \end{align*}

Problem 12110

ODE	\begin {align*} x^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=-x \left (t \right )+y \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x \left (t \right ) &= c_{1} t +c_{2} \\ y \left (t \right ) &= c_{1} t +c_{1} +c_{2} \\ \end{align*}

Problem 12111

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

Problem 12112

ODE	\[ \boxed {6 y+\left (5 x +2 y-3\right ) y^{\prime }=-12 x +9} \]
program solution	\[ \frac {2 \ln \left (3 x +2 y-3\right )}{5}+\frac {3 \ln \left (8 x +2 y-3\right )}{5} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\operatorname {RootOf}\left (128 \textit {\_Z}^{25} c_{1} x^{5}+640 \textit {\_Z}^{20} c_{1} x^{5}+800 \textit {\_Z}^{15} c_{1} x^{5}-1\right )^{5} x -4 x +\frac {3}{2} \]

Problem 12113

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

Problem 12114

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

Problem 12115

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

Problem 12116

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

Problem 12117

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

Problem 12118

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

Problem 12119

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

Problem 12120

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

Problem 12121

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

Problem 12122

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

Problem 12123

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

Problem 12124

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

Problem 12125

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

Problem 12126

ODE	\[ \boxed {-{y^{\prime }}^{3}+y^{\prime }=-x +2} \]
program solution	\[ y = \int \frac {\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}}+12}{6 \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {1}{3}}}d x +c_{1} \] Verified OK. \[ y = \int \frac {i \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}-\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}}-12}{12 \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {1}{3}}}d x +c_{2} \] Verified OK. \[ y = \int -\frac {i \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}+\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}}+12}{12 \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {1}{3}}}d x +c_{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (\int \frac {i \sqrt {3}\, \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}}-12 i \sqrt {3}+\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}}+12}{\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (i \sqrt {3}-1\right ) \left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}}-12 i \sqrt {3}-12}{\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {2}{3}}+12}{\left (-216+108 x +12 \sqrt {81 x^{2}-324 x +312}\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align*}

Problem 12127

ODE	\[ \boxed {y^{\prime }-\frac {y}{x +y^{3}}=0} \]
program solution	\[ \frac {y^{2}}{2}-\frac {x}{y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}-6 c_{1}}{3 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 i \sqrt {3}\, c_{1} +\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}-6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 i \sqrt {3}\, c_{1} -\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {2}{3}}+6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 12128

ODE	\[ \boxed {y-{y^{\prime }}^{4}+{y^{\prime }}^{3}=-2} \]
program solution	\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-\textit {\_Z}^{3}-\textit {\_a} -2\right )}d \textit {\_a} = x +c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -2 \\ y \left (x \right ) &= \frac {12 \left (\frac {243}{16384}+\frac {\left (\frac {9}{64}-c_{1} +x \right ) \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}}{16}+\frac {c_{1}^{2}}{2}+\left (-\frac {9}{64}-x \right ) c_{1} +\frac {x^{2}}{2}+\frac {9 x}{64}\right ) \left (27-192 c_{1} +192 x +24 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )^{\frac {2}{3}}+24 \left (x -c_{1} -\frac {7949}{1536}\right ) \left (\frac {9}{64}-c_{1} +x +\frac {\sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}}{8}\right ) \left (27-192 c_{1} +192 x +24 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )^{\frac {1}{3}}+\frac {27 \left (\frac {9}{64}-c_{1} +x \right ) \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}}{2}+108 \left (x -c_{1} +\frac {9}{128}\right ) \left (x -c_{1} +\frac {27}{128}\right )}{\left (27-192 c_{1} +192 x +24 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )^{\frac {1}{3}} \left (9-64 c_{1} +64 x +8 \sqrt {64}\, \sqrt {\left (x -c_{1} +\frac {9}{32}\right ) \left (x -c_{1} \right )}\right )} \\ \end{align*}

Problem 12129

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

Problem 12130

ODE	\[ \boxed {y^{\prime }-\frac {2 y-x -4}{2 x -y+5}=0} \]
program solution	\[ -\frac {\ln \left (y-3-x \right )}{2}+\frac {3 \ln \left (x +y+1\right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (i \sqrt {3}-1\right ) \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (x +2\right )^{2}-1}+27 c_{1} \left (x +2\right )\right )^{\frac {2}{3}}-3 i \sqrt {3}-3+6 \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (x +2\right )^{2}-1}+27 c_{1} x +54 c_{1} \right )^{\frac {1}{3}} \left (1+x \right ) c_{1}}{6 \left (3 \sqrt {3}\, \sqrt {27 c_{1}^{2} \left (x +2\right )^{2}-1}+27 c_{1} \left (x +2\right )\right )^{\frac {1}{3}} c_{1}} \]

Problem 12131

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

Problem 12132

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

Problem 12133

ODE	\[ \boxed {y^{\prime }-y^{3} x=x^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 12134

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

Problem 12135

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

Problem 12136

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

Problem 12137

ODE	\[ \boxed {y-5 y^{\prime } x +{y^{\prime }}^{2}=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {5 x}{9}+\frac {\sqrt {25 x^{2}-4 y}}{9}+\frac {4 c_{1} \sqrt {2}}{\left (10 x +2 \sqrt {25 x^{2}-4 y}\right )^{\frac {5}{4}}} \] Verified OK. \[ x = \frac {5 x}{9}-\frac {\sqrt {25 x^{2}-4 y}}{9}+\frac {4 c_{1} \sqrt {2}}{\left (10 x -2 \sqrt {25 x^{2}-4 y}\right )^{\frac {5}{4}}} \] Verified OK.
Maple solution	\begin{align*} -\frac {4 \sqrt {2}\, c_{1}}{\left (10 x -2 \sqrt {25 x^{2}-4 y \left (x \right )}\right )^{\frac {5}{4}}}+\frac {4 x}{9}+\frac {\sqrt {25 x^{2}-4 y \left (x \right )}}{9} &= 0 \\ -\frac {4 \sqrt {2}\, c_{1}}{\left (10 x +2 \sqrt {25 x^{2}-4 y \left (x \right )}\right )^{\frac {5}{4}}}+\frac {4 x}{9}-\frac {\sqrt {25 x^{2}-4 y \left (x \right )}}{9} &= 0 \\ \end{align*}

Problem 12138

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

Problem 12139

ODE	\[ \boxed {y^{\prime }-\left (x -5 y\right )^{\frac {1}{3}}=2} \]
program solution	\[ \frac {81 \ln \left (729+125 x -625 y\right )}{625}-\frac {27 \left (x -5 y\right )^{\frac {1}{3}}}{125}+\frac {162 \ln \left (5 \left (x -5 y\right )^{\frac {1}{3}}+9\right )}{625}-\frac {81 \ln \left (25 \left (x -5 y\right )^{\frac {2}{3}}-45 \left (x -5 y\right )^{\frac {1}{3}}+81\right )}{625}+\frac {3 \left (x -5 y\right )^{\frac {2}{3}}}{50} = -\frac {x}{5}+c_{1} \] Verified OK.
Maple solution	\[ x +\frac {81 \ln \left (729-625 y \left (x \right )+125 x \right )}{125}-\frac {27 \left (x -5 y \left (x \right )\right )^{\frac {1}{3}}}{25}-\frac {81 \ln \left (25 \left (x -5 y \left (x \right )\right )^{\frac {2}{3}}-45 \left (x -5 y \left (x \right )\right )^{\frac {1}{3}}+81\right )}{125}+\frac {162 \ln \left (9+5 \left (x -5 y \left (x \right )\right )^{\frac {1}{3}}\right )}{125}+\frac {3 \left (x -5 y \left (x \right )\right )^{\frac {2}{3}}}{10}-c_{1} = 0 \]

Problem 12140

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

Problem 12141

ODE	\[ \boxed {x^{\prime }+5 x=10 t +2} \] With initial conditions \begin {align*} [x \left (1\right ) = 2] \end {align*}
program solution	\[ x = 2 t \] Verified OK.
Maple solution	\[ x \left (t \right ) = 2 t \]

Problem 12142

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

Problem 12143

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

Problem 12144

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

Problem 12145

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

Problem 12146

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

Problem 12147

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

Problem 12148

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

Problem 12149

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

Problem 12150

ODE	\[ \boxed {-y+\left (x^{2} y^{2}+x \right ) y^{\prime }=-x^{2}} \]
program solution	\[ x +\frac {y}{x}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \left (-\frac {2^{\frac {1}{3}} {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}}{2}+x \right )}{{\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+2 i \sqrt {3}\, x -2 x \right ) 2^{\frac {1}{3}}}{4 {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}} {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {2}{3}}+2 \left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} x}{4 {\left (\left (-3 c_{1} x -3 x^{2}+\sqrt {\frac {9 c_{1}^{2} x^{3}+18 x^{4} c_{1} +9 x^{5}+4}{x}}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}

Problem 12151

ODE	\[ \boxed {3 y^{2}+2 y \left (y^{2}-3 x \right ) y^{\prime }=x} \]
program solution	\[ \ln \left (x +y^{2}\right )-\frac {\ln \left (-x +y^{2}\right )}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2 c_{1} -2 \sqrt {c_{1} \left (c_{1} -8 x \right )}-4 x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2 c_{1} -2 \sqrt {c_{1} \left (c_{1} -8 x \right )}-4 x}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {2 c_{1} +2 \sqrt {c_{1} \left (c_{1} -8 x \right )}-4 x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2 c_{1} +2 \sqrt {c_{1} \left (c_{1} -8 x \right )}-4 x}}{2} \\ \end{align*}

Problem 12152

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

Problem 12153

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

Problem 12154

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

Problem 12155

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

Problem 12156

ODE	\[ \boxed {\left (4 y+2 x +3\right ) y^{\prime }-2 y=x +1} \]
program solution	\[ y = \frac {\operatorname {LambertW}\left ({\mathrm e}^{8 x +5+16 c_{1}}\right )}{8}-\frac {x}{2}-\frac {5}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{2}+\frac {\operatorname {LambertW}\left (c_{1} {\mathrm e}^{5+8 x}\right )}{8}-\frac {5}{8} \]

Problem 12157

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

Problem 12158

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

Problem 12159

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

Problem 12160

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

Problem 12161

ODE	\[ \boxed {{y^{\prime }}^{2}-2 y^{\prime } x +y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\left (8 x^{2}-2 y\right ) \sqrt {x^{2}-y}+8 x^{3}-6 x y+3 c_{1}}{3 \left (x +\sqrt {x^{2}-y}\right )^{2}} \] Verified OK. \[ x = \frac {\left (-8 x^{2}+2 y\right ) \sqrt {x^{2}-y}+8 x^{3}-6 x y+3 c_{1}}{3 \left (x -\sqrt {x^{2}-y}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (x^{2}+x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right ) \left (x^{2}-3 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right )}{4 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}-2 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right ) \left (i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}-i \sqrt {3}\, x^{2}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}+6 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+x^{2}\right )}{16 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}+x^{2}-2 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right ) \left (i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}+x^{2}+6 x \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {1}{3}}+\left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}\right )}{16 \left (x^{3}+2 \sqrt {3}\, \sqrt {-c_{1} \left (x^{3}-3 c_{1} \right )}-6 c_{1} \right )^{\frac {2}{3}}} \\ \end{align*}

Problem 12162

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

Problem 12163

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+10 y=100} \] With initial conditions \begin {align*} [y \left (0\right ) = 10, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = 10+5 \,{\mathrm e}^{3 x} \sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = 5 \,{\mathrm e}^{3 x} \sin \left (x \right )+10 \]

Problem 12164

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

Problem 12165

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

Problem 12166

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

Problem 12167

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

Problem 12168

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

Problem 12169

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

Problem 12170

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

Problem 12171

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

Problem 12172

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

Problem 12173

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-16 y=x^{2}-{\mathrm e}^{x}} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{-2 i x} c_{3} +{\mathrm e}^{2 i x} c_{4} +\frac {{\mathrm e}^{x}}{15}-\frac {x^{2}}{16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (\left (-16 c_{1} +\frac {1}{4}\right ) \cos \left (2 x \right )+x^{2}-16 c_{4} \sin \left (2 x \right )\right ) {\mathrm e}^{2 x}-16 c_{3} {\mathrm e}^{4 x}-16 c_{2} -\frac {16 \,{\mathrm e}^{3 x}}{15}\right ) {\mathrm e}^{-2 x}}{16} \]

Problem 12174

ODE
program solution
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {1}{2} x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= c_{2} +c_{1} x +\frac {1}{2} x^{2} \\ y \left (x \right ) &= c_{1} +c_{2} x +\sqrt {-c_{3}^{2}+1}\, \sin \left (x \right )+c_{3} \cos \left (x \right ) \\ \end{align*}

Problem 12175

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

Problem 12176

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

Problem 12177

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

Problem 12178

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (9 x^{2}-\frac {1}{25}\right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (-\frac {1}{5}, 3 x \right )+c_{2} \operatorname {BesselY}\left (-\frac {1}{5}, 3 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (\frac {1}{5}, 3 x \right )+c_{2} \operatorname {BesselY}\left (\frac {1}{5}, 3 x \right ) \]

Problem 12179

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

Problem 12180

ODE	\[ \boxed {y^{\prime \prime }-3 \sqrt {y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
program solution	\[ y = \frac {\left (x^{2}+4 x +4\right )^{2}}{16} \] Verified OK. \[ y = \frac {\left (x^{2}-4 x +4\right )^{2}}{16} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \frac {\left (x +2\right )^{4}}{16} \]

Problem 12181

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

Problem 12182

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

Problem 12183

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

Problem 12184

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

Problem 12185

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

Problem 12186

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

Problem 12187

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

Problem 12188

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

Problem 12189

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

Problem 12190

ODE	\[ \boxed {m x^{\prime \prime }-f \left (x\right )=0} \]
program solution
Maple solution	\begin{align*} m \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {m \left (c_{1} m +2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )\right )}}d \textit {\_b} \right )-t -c_{2} &= 0 \\ -m \left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {m \left (c_{1} m +2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )\right )}}d \textit {\_b} \right )-t -c_{2} &= 0 \\ \end{align*}

Problem 12191

ODE	\[ \boxed {m x^{\prime \prime }-f \left (x^{\prime }\right )=0} \]
program solution	\[ \int _{}^{x}\frac {1}{\operatorname {RootOf}\left (-\left (\int _{}^{\textit {\_Z}}\frac {m \textit {\_a}}{f \left (\textit {\_a} \right )}d \textit {\_a} \right )+\textit {\_a} +c_{1} \right )}d \textit {\_a} = t +c_{2} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \int \operatorname {RootOf}\left (t -m \left (\int _{}^{\textit {\_Z}}\frac {1}{f \left (\textit {\_f} \right )}d \textit {\_f} \right )+c_{1} \right )d t +c_{2} \]

Problem 12192

ODE	\[ \boxed {y^{\left (6\right )}-3 y^{\left (5\right )}+3 y^{\prime \prime \prime \prime }-y^{\prime \prime \prime }=x} \]
program solution	\[ y = \left (c_{6} x^{2}+c_{5} x +c_{4} \right ) {\mathrm e}^{x}+c_{3} x^{2}+c_{2} x +c_{1} -\frac {x^{4}}{24}-\frac {x^{3}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{3} x^{2}+\left (c_{2} -6 c_{3} \right ) x +c_{1} -3 c_{2} +12 c_{3} \right ) {\mathrm e}^{x}-\frac {x^{4}}{24}-\frac {x^{3}}{2}+\frac {c_{4} x^{2}}{2}+c_{5} x +c_{6} \]

Problem 12193

ODE	\[ \boxed {x^{\prime \prime \prime \prime }+2 x^{\prime \prime }+x=\cos \left (t \right )} \]
program solution	\[ x = \left (c_{2} t +c_{1} \right ) {\mathrm e}^{i t}+\left (c_{4} t +c_{3} \right ) {\mathrm e}^{-i t}+\frac {\cos \left (t \right ) \left (-4 t^{2}+2 i t +5\right )}{32}-\frac {\sin \left (t \right ) \left (i-6 t \right )}{32} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (8 c_{3} t -t^{2}+8 c_{1} +2\right ) \cos \left (t \right )}{8}+\left (\left (c_{4} +\frac {3}{8}\right ) t +c_{2} \right ) \sin \left (t \right ) \]

Problem 12194

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

Problem 12195

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

Problem 12196

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

Problem 12197

ODE	\[ \boxed {{y^{\prime \prime }}^{3}+y^{\prime \prime }=x -1} \]
program solution	\[ y = \int \int \frac {\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {2}{3}}-12}{6 \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {1}{3}}}d x d x +c_{1} x +c_{2} \] Verified OK. \[ y = \int \int -\frac {\left (1+i \sqrt {3}\right ) \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {2}{3}}+12 i \sqrt {3}-12}{12 \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {1}{3}}}d x d x +c_{3} x +c_{4} \] Verified OK. \[ y = \int \int \frac {\left (i \sqrt {3}-1\right ) \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12}{12 \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {1}{3}}}d x d x +c_{5} x +c_{6} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (\int \int \frac {\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {2}{3}}-12}{\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {1}{3}}}d x d x \right )}{6}+c_{1} x +c_{2} \\ y \left (x \right ) &= -\frac {\left (\int \int \frac {i \sqrt {3}\, \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {2}{3}}-12}{\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {1}{3}}}d x d x \right )}{12}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\left (\int \int \frac {\left (i \sqrt {3}-1\right ) \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12}{\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{\frac {1}{3}}}d x d x \right )}{12}+c_{1} x +c_{2} \\ \end{align*}

Problem 12198

ODE	\[ \boxed {x^{\prime \prime }+10 x^{\prime }+25 x=2^{t}+t \,{\mathrm e}^{-5 t}} \]
program solution	\[ x = {\mathrm e}^{-5 t} \left (c_{2} t +c_{1} \right )+\frac {t^{3} \left (\ln \left (2\right )+5\right )^{2} {\mathrm e}^{-5 t}+6 \,2^{t}}{6 \left (\ln \left (2\right )+5\right )^{2}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (\ln \left (2\right )+5\right )^{2} \left (t^{3}+6 c_{1} t +6 c_{2} \right ) {\mathrm e}^{-5 t}+6 \,2^{t}}{6 \left (\ln \left (2\right )+5\right )^{2}} \]

Problem 12199

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

Problem 12200

ODE	\[ \boxed {y^{\left (6\right )}-y={\mathrm e}^{2 x}} \]
program solution	\[ y = {\mathrm e}^{-x} c_{1} +{\mathrm e}^{x} c_{2} +{\mathrm e}^{\frac {\left (1+i \sqrt {3}\right ) x}{2}} c_{3} +{\mathrm e}^{-\frac {\left (1+i \sqrt {3}\right ) x}{2}} c_{4} +{\mathrm e}^{\frac {\left (i \sqrt {3}-1\right ) x}{2}} c_{5} +{\mathrm e}^{-\frac {\left (i \sqrt {3}-1\right ) x}{2}} c_{6} +\frac {{\mathrm e}^{2 x}}{63} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (\left (c_{3} {\mathrm e}^{\frac {x}{2}}+c_{5} {\mathrm e}^{\frac {3 x}{2}}\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\left ({\mathrm e}^{\frac {x}{2}} c_{4} +c_{6} {\mathrm e}^{\frac {3 x}{2}}\right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )+{\mathrm e}^{2 x} c_{1} +\frac {{\mathrm e}^{3 x}}{63}+c_{2} \right ) \]