2.17.62 Problems 6101 to 6200

Problem 6101

ODE	\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{1} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+y_{2} \left (x \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 1, y_{2} \left (0\right ) = 2] \]
program solution
Maple solution	\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{x} \\ y_{2} \left (x \right ) &= \left (x +2\right ) {\mathrm e}^{x} \\ \end{align*}

Problem 6102

ODE	\begin {align*} y_{1}^{\prime }\left (x \right )&=y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=6 y_{1} \left (x \right )+y_{2} \left (x \right ) \end {align*} With initial conditions \[ [y_{1} \left (0\right ) = 1, y_{2} \left (0\right ) = -1] \]
program solution
Maple solution	\begin{align*} y_{1} \left (x \right ) &= \frac {4 \,{\mathrm e}^{-2 x}}{5}+\frac {{\mathrm e}^{3 x}}{5} \\ y_{2} \left (x \right ) &= -\frac {8 \,{\mathrm e}^{-2 x}}{5}+\frac {3 \,{\mathrm e}^{3 x}}{5} \\ \end{align*}

Problem 6103

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

Problem 6104

ODE	\begin {align*} y_{1}^{\prime }\left (x \right )&=3 y_{1} \left (x \right )+x y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{2} \left (x \right )+x^{3} y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=2 x y_{2} \left (x \right )-y_{2} \left (x \right )+{\mathrm e}^{x} y_{3} \left (x \right ) \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6105

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

Problem 6106

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

Problem 6107

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

Problem 6108

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

Problem 6109

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

Problem 6110

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

Problem 6111

ODE	\[ \boxed {y^{\prime } x +y-y^{\prime } \sqrt {1-y x^{2}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6112

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

Problem 6113

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

Problem 6114

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

Problem 6115

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

Problem 6116

ODE	\[ \boxed {y^{\prime }-\frac {y^{2}}{y x -x^{2}}=0} \]
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 6117

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

Problem 6118

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

Problem 6119

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

Problem 6120

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

Problem 6121

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

Problem 6122

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

Problem 6123

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

Problem 6124

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

Problem 6125

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

Problem 6126

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

Problem 6127

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

Problem 6128

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

Problem 6129

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

Problem 6130

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

Problem 6131

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

Problem 6132

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

Problem 6133

ODE	\[ \boxed {x \left (x^{2}-4\right ) y^{\prime }=1} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = -\frac {\ln \left (x \right )}{4}+\frac {\ln \left (x^{2}-4\right )}{8}-\frac {\ln \left (3\right )}{8}-\frac {i \pi }{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x +2\right )}{8}+\frac {\ln \left (-2+x \right )}{8}-\frac {\ln \left (x \right )}{4}-\frac {\ln \left (3\right )}{8}-\frac {i \pi }{8} \]

Problem 6134

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

Problem 6135

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

Problem 6136

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

Problem 6137

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

Problem 6138

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

Problem 6139

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

Problem 6140

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

Problem 6141

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

Problem 6142

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

Problem 6143

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

Problem 6144

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

Problem 6145

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

Problem 6146

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

Problem 6147

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

Problem 6148

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

Problem 6149

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

Problem 6150

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

Problem 6151

ODE	\[ \boxed {\frac {y^{\prime }}{x^{2}+1}-\frac {x}{y}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}
program solution	\[ y = \frac {\sqrt {2 x^{4}+4 x^{2}+30}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {2 x^{4}+4 x^{2}+30}}{2} \]

Problem 6152

ODE	\[ \boxed {y^{\prime } y^{2}=x +2} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = \frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (12 x^{2}+48 x +512\right )^{\frac {1}{3}}}{2} \]

Problem 6153

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

Problem 6154

ODE	\[ \boxed {y^{\prime } \left (y+1\right )=-x^{2}+1} \] With initial conditions \begin {align*} [y \left (-1\right ) = -2] \end {align*}
program solution	\[ y = -1-\frac {\sqrt {-6 x^{3}+18 x +21}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -1-\frac {\sqrt {-6 x^{3}+18 x +21}}{3} \]

Problem 6155

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

Problem 6156

ODE	\[ \boxed {y^{\prime \prime } y^{\prime }=x \left (1+x \right )} \]
program solution	\[ y = \int \frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{2} \] Verified OK. \[ y = \int -\frac {\sqrt {6 x^{3}+9 x^{2}+18 c_{1}}}{3}d x +c_{3} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (\int \sqrt {6 x^{3}+9 x^{2}+9 c_{1}}d x \right )}{3}+c_{2} \\ y \left (x \right ) &= \frac {\left (\int \sqrt {6 x^{3}+9 x^{2}+9 c_{1}}d x \right )}{3}+c_{2} \\ \end{align*}

Problem 6157

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

Problem 6158

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

Problem 6159

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

Problem 6160

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

Problem 6161

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

Problem 6162

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

Problem 6163

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

Problem 6164

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

Problem 6165

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

Problem 6166

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

Problem 6167

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

Problem 6168

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

Problem 6169

ODE	\[ \boxed {x \ln \left (x \right ) y^{\prime }+y=3 x^{3}} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 6170

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

Problem 6171

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

Problem 6172

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

Problem 6173

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

Problem 6174

ODE	\[ \boxed {x y^{2} y^{\prime }+y^{3}=x \cos \left (x \right )} \]
program solution	\[ \frac {\left (-3 x^{3}+18 x \right ) \sin \left (x \right )}{3}+\frac {y^{3} x^{3}}{3}-3 x^{2} \cos \left (x \right )+6 \cos \left (x \right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {{\left (9 \left (x^{2}-2\right ) \cos \left (x \right )+3 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{\frac {1}{3}}}{x} \\ y \left (x \right ) &= -\frac {{\left (9 \left (x^{2}-2\right ) \cos \left (x \right )+3 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 x} \\ y \left (x \right ) &= \frac {{\left (9 \left (x^{2}-2\right ) \cos \left (x \right )+3 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2 x} \\ \end{align*}

Problem 6175

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

Problem 6176

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

Problem 6177

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

Problem 6178

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

Problem 6179

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

Problem 6180

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

Problem 6181

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

Problem 6182

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

Problem 6183

ODE	\[ \boxed {\sin \left (x \right ) \tan \left (y\right )+\cos \left (x \right ) \sec \left (x \right )^{2} y y^{\prime }=-1} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6184

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

Problem 6185

ODE	\[ \boxed {2 y^{2}-\left (4-2 y+4 y x \right ) y^{\prime }=4 x -5} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6186

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

Problem 6187

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

Problem 6188

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

Problem 6189

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

Problem 6190

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

Problem 6191

ODE	\[ \boxed {2 y^{3} x +y \cos \left (x \right )+\left (3 x^{2} y^{2}+\sin \left (x \right )\right ) y^{\prime }=0} \]
program solution	\[ y \left (x^{2} y^{2}+\sin \left (x \right )\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}-12 \sin \left (x \right )}{6 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}+12 i \sin \left (x \right ) \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}-12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}+12 i \sin \left (x \right ) \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {2}{3}}+12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{\frac {1}{3}}} \\ \end{align*}

Problem 6192

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

Problem 6193

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

Problem 6194

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

Problem 6195

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

Problem 6196

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

Problem 6197

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

Problem 6198

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

Problem 6199

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

Problem 6200

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