2.17.135 Problems 13401 to 13500

Problem 13401

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

Problem 13402

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

Problem 13403

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

Problem 13404

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

Problem 13405

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

Problem 13406

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

Problem 13407

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

Problem 13408

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

Problem 13409

ODE	\[ \boxed {4 y x^{3}+\left (x^{4}-y^{4}\right ) y^{\prime }=0} \]
program solution	\[ -\frac {y^{5}}{5}+x^{4} y = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-5 \textit {\_Z} \,c_{1}^{4} x^{4}+\textit {\_Z}^{5}-1\right )}{c_{1}} \]

Problem 13410

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

Problem 13411

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

Problem 13412

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

Problem 13413

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

Problem 13414

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

Problem 13415

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

Problem 13416

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

Problem 13417

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

Problem 13418

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

Problem 13419

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

Problem 13420

ODE	\[ \boxed {4 y x +\left (3 x^{2}+5 y\right ) y^{\prime }=0} \]
program solution	\[ 2 y^{\frac {3}{2}} x^{2}+2 y^{\frac {5}{2}} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (x^{5} \textit {\_Z}^{25}+x^{5} \textit {\_Z}^{15}-c_{1} \right )^{10} x^{2} \]

Problem 13421

ODE	\[ \boxed {12 y^{2} x^{2}+\left (7 y x^{3}+\frac {x}{y}\right ) y^{\prime }=-6} \]
program solution	\[ 5 \left (y x \right )^{\frac {1}{5}} \left (y^{2} x^{2}+1\right )^{\frac {3}{5}} x = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z}^{35} c_{1}^{2} x^{10}-\textit {\_Z}^{30} c_{1}^{2} x^{10}-1\right )^{15} x^{4} \left (\operatorname {RootOf}\left (\textit {\_Z}^{35} c_{1}^{2} x^{10}-\textit {\_Z}^{30} c_{1}^{2} x^{10}-1\right )^{5}-1\right ) c_{1} \]

Problem 13422

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

Problem 13423

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

Problem 13424

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

Problem 13425

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

Problem 13426

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

Problem 13427

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

Problem 13428

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

Problem 13429

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

Problem 13430

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

Problem 13431

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

Problem 13432

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

Problem 13433

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

Problem 13434

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

Problem 13435

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

Problem 13436

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

Problem 13437

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

Problem 13438

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

Problem 13439

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

Problem 13440

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

Problem 13441

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

Problem 13442

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

Problem 13443

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

Problem 13444

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

Problem 13445

ODE	\[ \boxed {y^{\prime }-\frac {2 y+x}{-y+2 x}=0} \]
program solution	\[ \frac {\ln \left (y^{2}+x^{2}\right )}{2}+2 \arctan \left (\frac {x}{y}\right ) = c_{1} \] 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 13446

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

Problem 13447

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

Problem 13448

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

Problem 13449

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

Problem 13450

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

Problem 13451

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

Problem 13452

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

Problem 13453

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

Problem 13454

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

Problem 13455

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

Problem 13456

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

Problem 13457

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

Problem 13458

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

Problem 13459

ODE	\[ \boxed {\left (y-x +3\right )^{2} \left (y^{\prime }-1\right )=1} \]
program solution	\[ -\frac {x \left (x^{2}+\left (-3 y-9\right ) x +3 y^{2}+18 y+30\right )}{3}+\frac {\left (y+3\right )^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \left (-3 c_{1} +3 x \right )^{\frac {1}{3}}+x -3 \\ y \left (x \right ) &= -\frac {\left (-3 c_{1} +3 x \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (-3 c_{1} +3 x \right )^{\frac {1}{3}}}{2}+x -3 \\ y \left (x \right ) &= -\frac {\left (-3 c_{1} +3 x \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (-3 c_{1} +3 x \right )^{\frac {1}{3}}}{2}+x -3 \\ \end{align*}

Problem 13460

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

Problem 13461

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

Problem 13462

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

Problem 13463

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

Problem 13464

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

Problem 13465

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

Problem 13466

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

Problem 13467

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

Problem 13468

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

Problem 13469

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

Problem 13470

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

Problem 13471

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

Problem 13472

ODE	\[ \boxed {x y^{\prime \prime }+4 y^{\prime }=18 x^{2}} \]
program solution	\[ y = \frac {\left (6 x^{3}+c_{1} \right )^{2}}{36 x^{3}}+\frac {c_{2}}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{3}-\frac {c_{1}}{3 x^{3}}+c_{2} \]

Problem 13473

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

Problem 13474

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

Problem 13475

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

Problem 13476

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

Problem 13477

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

Problem 13478

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

Problem 13479

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

Problem 13480

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 13481

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

Problem 13482

ODE	\[ \boxed {x y^{\prime \prime }-{y^{\prime }}^{2}=6 x^{5}} \]
program solution	\[ y = \int \frac {\sqrt {6}\, x^{\frac {5}{2}} \left (c_{3} \operatorname {BesselJ}\left (1, \frac {2 \sqrt {6}\, x^{\frac {5}{2}}}{5}\right )+\operatorname {BesselY}\left (1, \frac {2 \sqrt {6}\, x^{\frac {5}{2}}}{5}\right )\right )}{c_{3} \operatorname {BesselJ}\left (0, \frac {2 \sqrt {6}\, x^{\frac {5}{2}}}{5}\right )+\operatorname {BesselY}\left (0, \frac {2 \sqrt {6}\, x^{\frac {5}{2}}}{5}\right )}d x +c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {6}\, \left (\int \frac {x^{\frac {5}{2}} \left (\operatorname {BesselY}\left (1, \frac {2 x^{\frac {5}{2}} \sqrt {6}}{5}\right ) c_{1} +\operatorname {BesselJ}\left (1, \frac {2 x^{\frac {5}{2}} \sqrt {6}}{5}\right )\right )}{c_{1} \operatorname {BesselY}\left (0, \frac {2 x^{\frac {5}{2}} \sqrt {6}}{5}\right )+\operatorname {BesselJ}\left (0, \frac {2 x^{\frac {5}{2}} \sqrt {6}}{5}\right )}d x \right )+c_{2} \]

Problem 13483

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

Problem 13484

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

Problem 13485

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

Problem 13486

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

Problem 13487

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

Problem 13488

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

Problem 13489

ODE
program solution
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {1}{12} x^{4}+\frac {1}{3} c_{1} x^{3}+\frac {1}{2} c_{1}^{2} x^{2}+c_{2} x +c_{3} \\ \end{align*}

Problem 13490

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

Problem 13491

ODE	\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 5, y^{\prime }\left (0\right ) = 15] \end {align*}
program solution	\[ y = 5 \,{\mathrm e}^{3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 5 \,{\mathrm e}^{3 x} \]

Problem 13492

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

Problem 13493

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

Problem 13494

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

Problem 13495

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

Problem 13496

ODE	\[ \boxed {y^{2} y^{\prime \prime }+y^{\prime \prime }+2 y {y^{\prime }}^{2}=0} \]
program solution	\[ y = \frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}-\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}} \] Verified OK. \[ y = -\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK. \[ y = -\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{4}+\frac {1}{2 \left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}{2}+\frac {1}{\left (6 c_{1} c_{2} +6 c_{1} x +2 \sqrt {9 c_{1}^{2} c_{2}^{2}+18 c_{1}^{2} c_{2} x +9 c_{1}^{2} x^{2}+2}\right )^{\frac {1}{3}}}\right )}{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -i \\ y \left (x \right ) &= i \\ y \left (x \right ) &= \frac {\left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}}-4}{2 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}}+4 i \sqrt {3}-4}{4 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}} \sqrt {3}-\left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {2}{3}}+4 i \sqrt {3}+4}{4 \left (12 c_{1} x +12 c_{2} +4 \sqrt {9 c_{1}^{2} x^{2}+18 c_{1} c_{2} x +9 c_{2}^{2}+4}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 13497

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

Problem 13498

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

Problem 13499

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

Problem 13500

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