2.17.54 Problems 5301 to 5400

Problem 5301

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

Problem 5302

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

Problem 5303

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

Problem 5304

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

Problem 5305

ODE	\[ \boxed {y^{\prime } x +y-x^{3} y^{6}=0} \]
program solution	\[ \frac {5 x^{3} y^{5}-2}{10 x^{5} y^{5}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2^{\frac {1}{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{\frac {1}{5}}}{2 c_{1} x^{3}+5 x} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) 2^{\frac {1}{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{\frac {1}{5}}}{8 c_{1} x^{3}+20 x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right ) 2^{\frac {1}{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{\frac {1}{5}}}{8 c_{1} x^{3}+20 x} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) 2^{\frac {1}{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{\frac {1}{5}}}{8 c_{1} x^{3}+20 x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) 2^{\frac {1}{5}} \left (x^{2} \left (2 c_{1} x^{2}+5\right )^{4}\right )^{\frac {1}{5}}}{8 c_{1} x^{3}+20 x} \\ \end{align*}

Problem 5306

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

Problem 5307

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

Problem 5308

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

Problem 5309

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

Problem 5310

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

Problem 5311

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

Problem 5312

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

Problem 5313

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

Problem 5314

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

Problem 5315

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

Problem 5316

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

Problem 5317

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

Problem 5318

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

Problem 5319

ODE	\[ \boxed {4 y y^{\prime } x^{2}-3 x \left (3 y^{2}+2\right )-2 \left (3 y^{2}+2\right )^{3}=0} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6}\, \sqrt {\frac {-3 c_{1} x^{8}+\sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+1}{3 c_{1} x^{8}-1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {6}\, \sqrt {\frac {-3 c_{1} x^{8}+\sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+1}{3 c_{1} x^{8}-1}}}{3} \\ y \left (x \right ) &= -\frac {\sqrt {\frac {-18 c_{1} x^{8}-6 \sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+6}{3 c_{1} x^{8}-1}}}{3} \\ y \left (x \right ) &= \frac {\sqrt {\frac {-18 c_{1} x^{8}-6 \sqrt {-3 \left (c_{1} x^{8}-\frac {1}{3}\right ) c_{1} x^{9}}+6}{3 c_{1} x^{8}-1}}}{3} \\ \end{align*}

Problem 5320

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

Problem 5321

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

Problem 5322

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

Problem 5323

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

Problem 5324

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

Problem 5325

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

Problem 5326

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

Problem 5327

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

Problem 5328

ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}+3 y^{\prime } x -y=0} \]
program solution	\[ \ln \left (y\right )-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {9 x^{2}+4 y^{3}}}{3 x}\right )}{3} = c_{1} \] Verified OK. \[ \ln \left (y\right )+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {9 x^{2}+4 y^{3}}}{3 x}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {18^{\frac {1}{3}} \left (-x^{2}\right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \left (-x^{2}\right )^{\frac {1}{3}} \left (3 i 3^{\frac {1}{6}}+3^{\frac {2}{3}}\right )}{4} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} \left (-x^{2}\right )^{\frac {1}{3}} \left (-3^{\frac {2}{3}}+3 i 3^{\frac {1}{6}}\right )}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a}^{3}-3 \sqrt {4 \textit {\_a}^{3}+9}+9}{\textit {\_a} \left (4 \textit {\_a}^{3}+9\right )}d \textit {\_a} \right )+2 c_{1} \right ) x^{\frac {2}{3}} \\ \end{align*}

Problem 5329

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

Problem 5330

ODE	\[ \boxed {16 y^{3} {y^{\prime }}^{2}-4 y^{\prime } x +y=0} \]
program solution	\[ \frac {\ln \left (2\right )}{2}+\frac {\ln \left (x \right )}{2}+\frac {\ln \left (x +\sqrt {x^{2}-4 y^{4}}\right )}{2} = \frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK. \[ y = {\mathrm e}^{\frac {\ln \left (2\right )}{4}+\frac {\ln \left (-8 \,{\mathrm e}^{4 c_{1}} {\mathrm e}^{-2 c_{1}}+2 \,{\mathrm e}^{2 c_{1}} {\mathrm e}^{-2 c_{1}} x \right )}{4}+\frac {c_{1}}{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-x}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {x}\, \sqrt {2}}{2} \\ y \left (x \right ) &= \frac {\sqrt {x}\, \sqrt {2}}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}-\frac {4 \textit {\_a}^{4}-\sqrt {-4 \textit {\_a}^{4}+1}-1}{\textit {\_a} \left (4 \textit {\_a}^{4}-1\right )}d \textit {\_a} \right )+c_{1} \right ) \sqrt {x} \\ \end{align*}

Problem 5331

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

Problem 5332

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

Problem 5333

ODE	\[ \boxed {y-2 y^{\prime } x -y^{2} {y^{\prime }}^{3}=0} \]
program solution	\[ \frac {3 \ln \left (x \right )}{4} = \int _{}^{\frac {y}{x^{\frac {3}{4}}}}-\frac {9 \textit {\_a} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{\frac {1}{3}} 3^{\frac {1}{6}}}{9 \,3^{\frac {1}{6}} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{\frac {1}{3}} \textit {\_a}^{2}-2 \,36^{\frac {1}{3}} {\left (\left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{2}\right )}^{\frac {1}{3}}+8 \,18^{\frac {1}{3}}}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {2 \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= -\frac {2 i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {2 i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ \end{align*}

Problem 5334

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

Problem 5335

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

Problem 5336

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

Problem 5337

ODE	\[ \boxed {y {y^{\prime }}^{2}-y^{\prime } x +3 y=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = -i \sqrt {2}\, x \] Verified OK. \[ y = i \sqrt {2}\, x \] Verified OK. \[ y = \frac {{\operatorname {RootOf}\left (\left (c_{2}^{4}-x^{4}\right ) \textit {\_Z}^{20}+2 c_{2}^{4} \textit {\_Z}^{16}-2 c_{2}^{4} \textit {\_Z}^{12}-8 c_{2}^{4} \textit {\_Z}^{8}-7 c_{2}^{4} \textit {\_Z}^{4}-2 c_{2}^{4}\right )}^{10} x^{3}}{c_{2}^{2} {\left ({\operatorname {RootOf}\left (\left (c_{2}^{4}-x^{4}\right ) \textit {\_Z}^{20}+2 c_{2}^{4} \textit {\_Z}^{16}-2 c_{2}^{4} \textit {\_Z}^{12}-8 c_{2}^{4} \textit {\_Z}^{8}-7 c_{2}^{4} \textit {\_Z}^{4}-2 c_{2}^{4}\right )}^{4}+1\right )}^{2} \left (\frac {{\operatorname {RootOf}\left (\left (c_{2}^{4}-x^{4}\right ) \textit {\_Z}^{20}+2 c_{2}^{4} \textit {\_Z}^{16}-2 c_{2}^{4} \textit {\_Z}^{12}-8 c_{2}^{4} \textit {\_Z}^{8}-7 c_{2}^{4} \textit {\_Z}^{4}-2 c_{2}^{4}\right )}^{20} x^{4}}{c_{2}^{4} {\left ({\operatorname {RootOf}\left (\left (c_{2}^{4}-x^{4}\right ) \textit {\_Z}^{20}+2 c_{2}^{4} \textit {\_Z}^{16}-2 c_{2}^{4} \textit {\_Z}^{12}-8 c_{2}^{4} \textit {\_Z}^{8}-7 c_{2}^{4} \textit {\_Z}^{4}-2 c_{2}^{4}\right )}^{4}+1\right )}^{4}}+3\right )} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ \ln \left (x \right )-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {-x^{2}+12 y \left (x \right )^{2}}{x^{2}}}}\right )}{4}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-12 y \left (x \right )^{2}}{x^{2}}}}{5}\right )}{4}+\frac {5 \ln \left (\frac {2 x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{8}-\frac {\ln \left (\frac {y \left (x \right )}{x}\right )}{4}-c_{1} &= 0 \\ \ln \left (x \right )+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {-x^{2}+12 y \left (x \right )^{2}}{x^{2}}}}\right )}{4}-\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-12 y \left (x \right )^{2}}{x^{2}}}}{5}\right )}{4}+\frac {5 \ln \left (\frac {2 x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{8}-\frac {\ln \left (\frac {y \left (x \right )}{x}\right )}{4}-c_{1} &= 0 \\ \end{align*}

Problem 5338

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

Problem 5339

ODE	\[ \boxed {y^{2} {y^{\prime }}^{2}+3 y^{\prime } x -y=0} \]
program solution	\[ \ln \left (y\right )-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {9 x^{2}+4 y^{3}}}{3 x}\right )}{3} = c_{1} \] Verified OK. \[ \ln \left (y\right )+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {9 x^{2}+4 y^{3}}}{3 x}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {18^{\frac {1}{3}} \left (-x^{2}\right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \left (-x^{2}\right )^{\frac {1}{3}} \left (3 i 3^{\frac {1}{6}}+3^{\frac {2}{3}}\right )}{4} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \left (-x^{2}\right )^{\frac {1}{3}} \left (3^{\frac {2}{3}}-3 i 3^{\frac {1}{6}}\right )}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {4 \textit {\_a}^{3}+9}+9}{\textit {\_a} \left (4 \textit {\_a}^{3}+9\right )}d \textit {\_a} \right )+2 c_{1} \right ) x^{\frac {2}{3}} \\ \end{align*}

Problem 5340

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

Problem 5341

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

Problem 5342

ODE	\[ \boxed {\left (3 y-1\right )^{2} {y^{\prime }}^{2}-4 y=0} \]
program solution	\[ \sqrt {y}\, \left (y-1\right ) = x +c_{1} \] Verified OK. \[ -\sqrt {y}\, \left (y-1\right ) = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {{\left (\left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12\right )}^{2}}{36 \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {{\left (i \sqrt {3}\, \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}-12 i \sqrt {3}+\left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12\right )}^{2}}{144 \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {{\left (\left (i \sqrt {3}-1\right ) \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}-12 i \sqrt {3}-12\right )}^{2}}{144 \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {{\left (\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12\right )}^{2}}{36 \left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {{\left (i \sqrt {3}\, \left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}-12 i \sqrt {3}+\left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12\right )}^{2}}{144 \left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {{\left (\left (i \sqrt {3}-1\right ) \left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}-12 i \sqrt {3}-12\right )}^{2}}{144 \left (108 x -108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ \end{align*}

Problem 5343

ODE	\[ \boxed {y+y^{\prime } x -x^{4} {y^{\prime }}^{2}=0} \]
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}{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}{x \,c_{1}^{2}} \\ y \left (x \right ) &= \frac {-c_{1} i-x}{x \,c_{1}^{2}} \\ \end{align*}

Problem 5344

ODE	\[ \boxed {2 y-{y^{\prime }}^{2}-4 y^{\prime } x=0} \]
program solution	\[ y = 0 \] Verified OK. \[ x = \frac {\left (-16 x^{2}-2 y\right ) \sqrt {4 x^{2}+2 y}+32 x^{3}+12 y x +3 c_{1}}{3 \left (2 x -\sqrt {4 x^{2}+2 y}\right )^{2}} \] Verified OK. \[ x = \frac {2 \left (8 x^{2}+y\right ) \sqrt {4 x^{2}+2 y}+32 x^{3}+12 y x +3 c_{1}}{3 \left (2 x +\sqrt {4 x^{2}+2 y}\right )^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-2 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right ) \left (\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+6 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right )}{8 \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right ) \left (i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x^{2}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}-12 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{2}\right )}{32 \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {\left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x^{2}+4 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}\right ) \left (4 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x^{2}-12 x \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}+\left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}\right )}{32 \left (12 c_{1} -8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {2}{3}}} \\ \end{align*}

Problem 5345

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

Problem 5346

ODE	\[ \boxed {{y^{\prime }}^{3}-4 y^{\prime } x^{4}+8 y x^{3}=0} \]
program solution	\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x^{3}}}\frac {3 \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}}{2 \,18^{\frac {1}{3}} {\left (\left (\sqrt {81 \textit {\_a}^{2}-12}-9 \textit {\_a} \right )^{2}\right )}^{\frac {1}{3}}-9 \textit {\_a} \left (12 \sqrt {81 \textit {\_a}^{2}-12}-108 \textit {\_a} \right )^{\frac {1}{3}}+12}d \textit {\_a} +c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {3}\, x^{3}}{9} \\ y \left (x \right ) &= \frac {2 \sqrt {3}\, x^{3}}{9} \\ y \left (x \right ) &= \frac {x^{2}}{2 c_{1}}-\frac {1}{8 c_{1}^{3}} \\ \end{align*}

Problem 5347

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

Problem 5348

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	\[ y \left (x \right ) = \left (c_{1} {\mathrm e}^{5 x}+c_{2} \right ) {\mathrm e}^{-3 x} \]

Problem 5349

ODE	\[ \boxed {y^{\prime \prime \prime }-6 y^{\prime \prime }+12 y^{\prime }-8 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{2 x}+c_{2} x \,{\mathrm e}^{2 x}+x^{2} {\mathrm e}^{2 x} c_{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{3} x^{2}+c_{2} x +c_{1} \right ) \]

Problem 5350

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

Problem 5351

ODE	\[ \boxed {y^{\prime \prime }+9 y=x \cos \left (x \right )} \]
program solution	\[ y = c_{1} \cos \left (3 x \right )+\frac {c_{2} \sin \left (3 x \right )}{3}+\frac {x \cos \left (x \right )}{8}+\frac {\sin \left (x \right )}{32} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (3 x \right ) c_{2} +\cos \left (3 x \right ) c_{1} +\frac {\sin \left (x \right )}{32}+\frac {\cos \left (x \right ) x}{8} \]

Problem 5352

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 5353

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

Problem 5354

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

Problem 5355

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

Problem 5356

ODE	\[ \boxed {y y^{\prime \prime }+{y^{\prime }}^{2}=2} \]
program solution	\[ \frac {y^{2}}{2}-\frac {c_{1} x}{2}-\frac {x^{2}}{2}-c_{2} = 0 \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= \sqrt {-2 c_{1} x +2 x^{2}+2 c_{2}} \\ y \left (x \right ) &= -\sqrt {-2 c_{1} x +2 x^{2}+2 c_{2}} \\ \end{align*}

Problem 5357

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

Problem 5358

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-15 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 x}+\frac {c_{2} {\mathrm e}^{3 x}}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left ({\mathrm e}^{8 x} c_{2} +c_{1} \right ) {\mathrm e}^{-5 x} \]

Problem 5359

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

Problem 5360

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

Problem 5361

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+12 y^{\prime \prime }-8 y^{\prime }=0} \]
program solution	\[ y = c_{1} +c_{2} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} c_{3} +x^{2} {\mathrm e}^{2 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{4} x^{2}+c_{3} x +c_{2} \right ) {\mathrm e}^{2 x}+c_{1} \]

Problem 5362

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

Problem 5363

ODE	\[ \boxed {y^{\prime \prime }+25 y=0} \]
program solution	\[ y = c_{1} \cos \left (5 x \right )+\frac {c_{2} \sin \left (5 x \right )}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (5 x \right )+c_{2} \cos \left (5 x \right ) \]

Problem 5364

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

Problem 5365

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

Problem 5366

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime \prime }+13 y^{\prime \prime }-12 y^{\prime }+4 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} x \,{\mathrm e}^{x}+{\mathrm e}^{2 x} c_{3} +x \,{\mathrm e}^{2 x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} x +c_{1} \right ) {\mathrm e}^{2 x}+{\mathrm e}^{x} \left (c_{4} x +c_{3} \right ) \]

Problem 5367

ODE	\[ \boxed {y^{\left (6\right )}+9 y^{\prime \prime \prime \prime }+24 y^{\prime \prime }+16 y=0} \]
program solution	\[ y = {\mathrm e}^{i x} c_{1} +{\mathrm e}^{-i x} c_{2} +{\mathrm e}^{2 i x} c_{3} +x \,{\mathrm e}^{2 i x} c_{4} +{\mathrm e}^{-2 i x} c_{5} +x \,{\mathrm e}^{-2 i x} c_{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{6} x +c_{4} \right ) \cos \left (2 x \right )+\left (c_{5} x +c_{3} \right ) \sin \left (2 x \right )+c_{1} \sin \left (x \right )+\cos \left (x \right ) c_{2} \]

Problem 5368

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

Problem 5369

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

Problem 5370

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

Problem 5371

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

Problem 5372

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

Problem 5373

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

Problem 5374

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

Problem 5375

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

Problem 5376

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

Problem 5377

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

Problem 5378

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

Problem 5379

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

Problem 5380

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

Problem 5381

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

Problem 5382

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

Problem 5383

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

Problem 5384

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

Problem 5385

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

Problem 5386

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

Problem 5387

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

Problem 5388

ODE	\[ \boxed {y^{\prime \prime }-9 y=x +{\mathrm e}^{2 x}-\sin \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 x}+\frac {c_{2} {\mathrm e}^{3 x}}{6}-\frac {{\mathrm e}^{2 x}}{5}-\frac {x}{9}+\frac {\sin \left (2 x \right )}{13} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {{\mathrm e}^{-3 x} \left (\left (x -\frac {9 \sin \left (2 x \right )}{13}\right ) {\mathrm e}^{3 x}-9 \,{\mathrm e}^{6 x} c_{1} -9 c_{2} +\frac {9 \,{\mathrm e}^{5 x}}{5}\right )}{9} \]

Problem 5389

ODE	\[ \boxed {y^{\prime \prime \prime }+3 y^{\prime \prime }+2 y^{\prime }=x^{2}+4 x +8} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} +{\mathrm e}^{-2 x} c_{3} +\frac {x^{3}}{6}+\frac {x^{2}}{4}+\frac {11 x}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2}}{4}+\frac {x^{3}}{6}-c_{2} {\mathrm e}^{-x}+\frac {{\mathrm e}^{-2 x} c_{1}}{2}+\frac {11 x}{4}+c_{3} \]

Problem 5390

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

Problem 5391

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

Problem 5392

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }+y={\mathrm e}^{3 x}+6 \,{\mathrm e}^{x}-3 \,{\mathrm e}^{-2 x}+5} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {2 \sqrt {3}\, {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_{2}}{3}+\frac {{\mathrm e}^{-2 x} \left ({\mathrm e}^{5 x}+65 \,{\mathrm e}^{2 x}+26 \,{\mathrm e}^{3 x}-13\right )}{13} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {{\mathrm e}^{5 x}}{13}+2 \,{\mathrm e}^{3 x}+\sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_{2} {\mathrm e}^{\frac {3 x}{2}}+\cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_{1} {\mathrm e}^{\frac {3 x}{2}}+5 \,{\mathrm e}^{2 x}-1\right ) {\mathrm e}^{-2 x} \]

Problem 5393

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

Problem 5394

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

Problem 5395

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

Problem 5396

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

Problem 5397

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

Problem 5398

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

Problem 5399

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

Problem 5400

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