2.17.63 Problems 6201 to 6300

Problem 6201

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

Problem 6202

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

Problem 6203

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

Problem 6204

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

Problem 6205

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

Problem 6206

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

Problem 6207

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

Problem 6208

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

Problem 6209

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

Problem 6210

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

Problem 6211

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

Problem 6212

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

Problem 6213

ODE	\[ \boxed {y^{\prime }-\frac {x +y+4}{x -y-6}=0} \]
program solution	\[ \frac {\ln \left (y^{2}+x^{2}+10 y-2 x +26\right )}{2}-\arctan \left (\frac {y+5}{x -1}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -5-\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -1\right )+2 c_{1} \right )\right ) \left (x -1\right ) \]

Problem 6214

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

Problem 6215

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

Problem 6216

ODE	\[ \boxed {y^{\prime }-\frac {x +y-1}{x +4 y+2}=0} \]
program solution	\[ \frac {3 \ln \left (2 y+4-x \right )}{4}+\frac {\ln \left (x +2 y\right )}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x -4\right ) \operatorname {RootOf}\left (\textit {\_Z}^{16}+\left (2 c_{1} x^{4}-16 c_{1} x^{3}+48 c_{1} x^{2}-64 c_{1} x +32 c_{1} \right ) \textit {\_Z}^{4}-c_{1} x^{4}+8 c_{1} x^{3}-24 c_{1} x^{2}+32 c_{1} x -16 c_{1} \right )^{4}-x +2}{2 \operatorname {RootOf}\left (\textit {\_Z}^{16}+\left (2 c_{1} x^{4}-16 c_{1} x^{3}+48 c_{1} x^{2}-64 c_{1} x +32 c_{1} \right ) \textit {\_Z}^{4}-c_{1} x^{4}+8 c_{1} x^{3}-24 c_{1} x^{2}+32 c_{1} x -16 c_{1} \right )^{4}} \]

Problem 6217

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

Problem 6218

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

Problem 6219

ODE	\[ \boxed {y^{\prime }-\frac {2+3 y^{2} x}{4 y x^{2}}=0} \]
program solution	\[ \frac {2 y^{2} x +\frac {4}{5}}{x^{\frac {5}{2}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {5}\, \sqrt {5 x^{\frac {7}{2}} c_{1} -2 x}}{5 x} \\ y \left (x \right ) &= \frac {\sqrt {5}\, \sqrt {5 x^{\frac {7}{2}} c_{1} -2 x}}{5 x} \\ \end{align*}

Problem 6220

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

Problem 6221

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

Problem 6222

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

Problem 6223

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}-y x}{y^{2} \cos \left (\frac {x}{y}\right )}=0} \]
program solution	\[ \int _{}^{\frac {y}{x}}\frac {\cos \left (\frac {1}{\textit {\_a}}\right ) \textit {\_a}^{2}}{\textit {\_a}^{3} \cos \left (\frac {1}{\textit {\_a}}\right )+\textit {\_a} -1}d \textit {\_a} +\ln \left (x \right )-c_{2} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2} \cos \left (\frac {1}{\textit {\_a}}\right )}{\textit {\_a}^{3} \cos \left (\frac {1}{\textit {\_a}}\right )+\textit {\_a} -1}d \textit {\_a} +\ln \left (x \right )+c_{1} \right ) x \]

Problem 6224

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

Problem 6225

ODE	\[ \boxed {\left (3 x^{2}-y^{2}\right ) y^{\prime }-2 y x=0} \]
program solution	\[ -\frac {x^{2}}{y^{3}}+\frac {1}{y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}}{3 c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 6226

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

Problem 6227

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

Problem 6228

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

Problem 6229

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

Problem 6230

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

Problem 6231

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

Problem 6232

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

Problem 6233

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

Problem 6234

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

Problem 6235

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

Problem 6236

ODE	\[ \boxed {y^{\prime }-\frac {2 y}{x}-\frac {x^{3}}{y}-x \tan \left (\frac {y}{x^{2}}\right )=0} \]
program solution	\[ -\ln \left (\cos \left (\frac {y}{x^{2}}\right ) x^{2}+\sin \left (\frac {y}{x^{2}}\right ) y\right )+2 \ln \left (x \right ) = -\ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2} \left (\csc \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) x -\cot \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) c_{1} \right )}{c_{1}} \\ y \left (x \right ) &= \frac {\left (\cot \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) c_{1} +\csc \left (\operatorname {RootOf}\left (2 c_{1}^{2} \sin \left (\textit {\_Z} \right )^{2} \textit {\_Z}^{2}-2 c_{1}^{2} \cos \left (\textit {\_Z} \right )^{2}-4 c_{1} \sin \left (\textit {\_Z} \right ) x \textit {\_Z} +2 x^{2}\right )\right ) x \right ) x^{2}}{c_{1}} \\ \end{align*}

Problem 6237

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

Problem 6238

ODE	\[ \boxed {x y y^{\prime \prime }-y^{\prime }-{y^{\prime }}^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 6239

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

Problem 6240

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

Problem 6241

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

Problem 6242

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

Problem 6243

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

Problem 6244

ODE	\[ \boxed {\left (x^{2}+2 y^{\prime }\right ) y^{\prime \prime }+2 y^{\prime } x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = -\frac {\left (\munderset {c_{1} \rightarrow 0}{\operatorname {lim}}\left (-\frac {4 \left (\left (1+i\right ) \operatorname {EllipticF}\left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x}{c_{1}^{\frac {1}{4}}}, i\right ) c_{1}^{\frac {5}{4}} \sqrt {\frac {i x^{2}+2 \sqrt {c_{1}}}{\sqrt {c_{1}}}}\, \sqrt {-\frac {i x^{2}-2 \sqrt {c_{1}}}{\sqrt {c_{1}}}}+\left (-\frac {x^{3}}{4}+\frac {3}{2}\right ) \sqrt {x^{4}+4 c_{1}}+\frac {x^{5}}{4}+c_{1} x \right )}{\sqrt {x^{4}+4 c_{1}}}\right )\right )}{6} \] Verified OK. \[ y = -\frac {\left (\munderset {c_{1} \rightarrow 0}{\operatorname {lim}}\frac {4 \left (1+i\right ) \operatorname {EllipticF}\left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x}{c_{1}^{\frac {1}{4}}}, i\right ) c_{1}^{\frac {5}{4}} \sqrt {\frac {i x^{2}+2 \sqrt {c_{1}}}{\sqrt {c_{1}}}}\, \sqrt {-\frac {i x^{2}-2 \sqrt {c_{1}}}{\sqrt {c_{1}}}}+4 \left (\frac {x^{3}}{4}-\frac {3}{2}\right ) \sqrt {x^{4}+4 c_{1}}+x^{5}+4 c_{1} x}{\sqrt {x^{4}+4 c_{1}}}\right )}{6} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 1 \\ y \left (x \right ) &= -\frac {x^{3}}{3}+1 \\ \end{align*}

Problem 6245

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

Problem 6246

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

Problem 6247

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

Problem 6248

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

Problem 6249

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

Problem 6250

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

Problem 6251

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

Problem 6252

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

Problem 6253

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

Problem 6254

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

Problem 6255

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

Problem 6256

ODE	\[ \boxed {-\sin \left (x \right ) \sin \left (y\right )+\cos \left (x \right ) \cos \left (y\right ) y^{\prime }=0} \]
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 6257

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

Problem 6258

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

Problem 6259

ODE	\[ \boxed {y^{\prime } y^{2}=x} \] With initial conditions \begin {align*} [y \left (-1\right ) = 3] \end {align*}
program solution	\[ \frac {y^{3}}{3}-\frac {x^{2}}{2} = {\frac {17}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (12 x^{2}+204\right )^{\frac {1}{3}}}{2} \]

Problem 6260

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

Problem 6261

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

Problem 6262

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

Problem 6263

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

Problem 6264

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

Problem 6265

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

Problem 6266

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

Problem 6267

ODE	\[ \boxed {y y^{\prime \prime }+y^{\prime }=0} \]
program solution	\[ \int _{}^{y}\frac {1}{-\ln \left (\textit {\_a} \right )+c_{1}}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= {\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{c_{1}} \operatorname {expIntegral}_{1}\left (-\textit {\_Z} +c_{1} \right )+x +c_{2} \right )} \\ \end{align*}

Problem 6268

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

Problem 6269

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

Problem 6270

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

Problem 6271

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

Problem 6272

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

Problem 6273

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

Problem 6274

ODE	\[ \boxed {y^{\prime \prime }-9 y^{\prime }+20 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{4 x}+c_{2} {\mathrm e}^{5 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{4 x} c_{1} +c_{2} {\mathrm e}^{5 x} \]

Problem 6275

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

Problem 6276

ODE	\[ \boxed {4 y^{\prime \prime }-12 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{\frac {3 x}{2}}+c_{2} x \,{\mathrm e}^{\frac {3 x}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {3 x}{2}} \left (c_{2} x +c_{1} \right ) \]

Problem 6277

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

Problem 6278

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

Problem 6279

ODE	\[ \boxed {4 y^{\prime \prime }+20 y^{\prime }+25 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {5 x}{2}}+c_{2} x \,{\mathrm e}^{-\frac {5 x}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {5 x}{2}} \left (c_{2} x +c_{1} \right ) \]

Problem 6280

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

Problem 6281

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 6282

ODE	\[ \boxed {4 y^{\prime \prime }-8 y^{\prime }+7 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) {\mathrm e}^{x} \sqrt {3}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{1} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+c_{2} \cos \left (\frac {\sqrt {3}\, x}{2}\right )\right ) \]

Problem 6283

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

Problem 6284

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

Problem 6285

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

Problem 6286

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

Problem 6287

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

Problem 6288

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+5 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 11] \end {align*}
program solution	\[ y = {\mathrm e}^{x}+2 \,{\mathrm e}^{5 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 \,{\mathrm e}^{5 x}+{\mathrm e}^{x} \]

Problem 6289

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = 5 x \,{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 5 x \,{\mathrm e}^{3 x} \]

Problem 6290

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

Problem 6291

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

Problem 6292

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }-9 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {9 \,{\mathrm e}^{x} {\mathrm e}^{-1}}{5}+\frac {{\mathrm e}^{-9 x} {\mathrm e}^{9}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {9 \,{\mathrm e}^{x -1}}{5}+\frac {{\mathrm e}^{9-9 x}}{5} \]

Problem 6293

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x +10 y=0} \]
program solution	\[ y = c_{1} x^{-1-3 i}-\frac {i c_{2} x^{-1+3 i}}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (3 \ln \left (x \right )\right )+c_{2} \cos \left (3 \ln \left (x \right )\right )}{x} \]

Problem 6294

ODE	\[ \boxed {2 x^{2} y^{\prime \prime }+10 y^{\prime } x +8 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{2}}+\frac {c_{2} \ln \left (x \right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \ln \left (x \right )+c_{1}}{x^{2}} \]

Problem 6295

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x -12 y=0} \]
program solution	\[ y = \frac {c_{1}}{x^{4}}+\frac {c_{2} x^{3}}{7} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} x^{7}+c_{2}}{x^{4}} \]

Problem 6296

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

Problem 6297

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

Problem 6298

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

Problem 6299

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x +3 y=0} \]
program solution	\[ y = c_{1} x^{-\frac {i \sqrt {11}}{2}-\frac {1}{2}}-\frac {i c_{2} \sqrt {11}\, x^{-\frac {1}{2}+\frac {i \sqrt {11}}{2}}}{11} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (\frac {\sqrt {11}\, \ln \left (x \right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {11}\, \ln \left (x \right )}{2}\right )}{\sqrt {x}} \]

Problem 6300

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