2.17.6 Problems 501 to 600

Problem 501

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

Problem 502

ODE	\[ \boxed {y^{\prime }-\frac {2-{\mathrm e}^{x}}{3+2 y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ -y^{2}-{\mathrm e}^{x}+2 x -3 y = -1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {3}{2}+\frac {\sqrt {13-4 \,{\mathrm e}^{x}+8 x}}{2} \]

Problem 503

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

Problem 504

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

Problem 505

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

Problem 506

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

Problem 507

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

Problem 508

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

Problem 509

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

Problem 510

ODE	\[ \boxed {y^{\prime }-\frac {4 y-3 x}{2 x -y}=0} \]
program solution	\[ -\frac {\ln \left (-x +y\right )}{4}+\frac {5 \ln \left (3 x +y\right )}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (-3+\operatorname {RootOf}\left (\textit {\_Z}^{20} c_{1} x^{4}-\textit {\_Z}^{4}+4\right )^{4}\right ) \]

Problem 511

ODE	\[ \boxed {y^{\prime }+\frac {4 x +3 y}{2 x +y}=0} \]
program solution	\[ \frac {2 \ln \left (4 x +y\right )}{3}+\frac {\ln \left (x +y\right )}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\frac {\left (-4 c_{1} x^{3}+\left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}\right )^{2}}{4 \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}} c_{1}}-x^{3}}{x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} +\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{6} c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} +\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}+4 x^{6} c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= -\frac {\frac {\left (4 \sqrt {3}\, c_{1} x^{3}+\sqrt {3}\, \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}+4 i c_{1} x^{3}-i \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}\right )^{2}}{16 \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}} c_{1}}+x^{3}}{x^{2}} \\ y \left (x \right ) &= -\frac {\frac {\left (4 \sqrt {3}\, c_{1} x^{3}+\sqrt {3}\, \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}-4 i c_{1} x^{3}+i \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}}\right )^{2}}{16 \left (4 c_{1} x^{3}+4 \sqrt {x^{6} c_{1}^{2} \left (4 c_{1} x^{3}+1\right )}\right )^{\frac {2}{3}} c_{1}}+x^{3}}{x^{2}} \\ y \left (x \right ) &= -\frac {2 \left (\frac {3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1}}{2}-\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (i \sqrt {3}-1\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+x^{6} \left (1+i \sqrt {3}\right ) c_{1}^{2}\right )}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} -\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (1+i \sqrt {3}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+2 x^{6} \left (i \sqrt {3}-1\right ) c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= -\frac {2 \left (\frac {3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1}}{2}-\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (i \sqrt {3}-1\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+x^{6} \left (1+i \sqrt {3}\right ) c_{1}^{2}\right )}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ y \left (x \right ) &= \frac {-3 x^{3} \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} -\frac {\left (c_{1} x^{3}+\sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right ) \left (1+i \sqrt {3}\right ) \left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {1}{3}}}{2}+2 x^{6} \left (i \sqrt {3}-1\right ) c_{1}^{2}}{\left (4 c_{1} x^{3}+4 \sqrt {4 c_{1}^{3} x^{9}+x^{6} c_{1}^{2}}\right )^{\frac {2}{3}} c_{1} x^{2}} \\ \end{align*}

Problem 512

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

Problem 513

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

Problem 514

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

Problem 515

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

Problem 516

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

Problem 517

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

Problem 518

ODE	\[ \boxed {\tan \left (t \right ) y+y^{\prime }=\sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 0] \end {align*}
program solution	\[ y = -i \cos \left (t \right ) \pi +\ln \left (\sec \left (t \right )\right ) \cos \left (t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (-\ln \left (\cos \left (t \right )\right )+i \pi \right ) \cos \left (t \right ) \]

Problem 519

ODE	\[ \boxed {2 y t +\left (-t^{2}+4\right ) y^{\prime }=3 t^{2}} \] With initial conditions \begin {align*} [y \left (-3\right ) = 1] \end {align*}
program solution	\[ y = \frac {3 \ln \left (2+t \right ) t^{2}}{8}-\frac {3 \ln \left (2+t \right )}{2}+\frac {11 t^{2}}{10}+\frac {3 t^{2} \ln \left (5\right )}{8}-\frac {3 \ln \left (t -2\right ) t^{2}}{8}+\frac {3 t}{2}-\frac {22}{5}-\frac {3 \ln \left (5\right )}{2}+\frac {3 \ln \left (t -2\right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {3 t}{2}+\frac {3 \ln \left (2+t \right ) t^{2}}{8}-\frac {3 \ln \left (2+t \right )}{2}-\frac {3 \ln \left (t -2\right ) t^{2}}{8}+\frac {3 \ln \left (t -2\right )}{2}+\frac {11 t^{2}}{10}-\frac {22}{5}+\frac {3 \ln \left (5\right ) t^{2}}{8}-\frac {3 \ln \left (5\right )}{2} \]

Problem 520

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

Problem 521

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

Problem 522

ODE	\[ \boxed {y^{\prime }-\frac {t^{2}+1}{3 y-y^{2}}=0} \]
program solution	\[ -\frac {t^{3}}{3}-\frac {y^{3}}{3}+\frac {3 y^{2}}{2}-t = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {\left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1} -162 t}\right )^{\frac {1}{3}}}{2}+\frac {9}{2 \left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4 t^{6}+24 c_{1} t^{3}+24 t^{4}-54 t^{3}+36 c_{1}^{2}+72 c_{1} t +36 t^{2}-162 c_{1} -162 t}\right )^{\frac {1}{3}}}+\frac {3}{2} \\ y \left (t \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 t +3 c_{1} -\frac {27}{2}\right ) \left (t^{3}+3 c_{1} +3 t \right )}\right )^{\frac {2}{3}}-9 i \sqrt {3}-6 \left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 t +3 c_{1} -\frac {27}{2}\right ) \left (t^{3}+3 c_{1} +3 t \right )}\right )^{\frac {1}{3}}+9}{4 \left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 t +3 c_{1} -\frac {27}{2}\right ) \left (t^{3}+3 c_{1} +3 t \right )}\right )^{\frac {1}{3}}} \\ y \left (t \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 t +3 c_{1} -\frac {27}{2}\right ) \left (t^{3}+3 c_{1} +3 t \right )}\right )^{\frac {2}{3}}-9 i \sqrt {3}+6 \left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 t +3 c_{1} -\frac {27}{2}\right ) \left (t^{3}+3 c_{1} +3 t \right )}\right )^{\frac {1}{3}}-9}{4 \left (27-4 t^{3}-12 c_{1} -12 t +2 \sqrt {4}\, \sqrt {\left (t^{3}+3 t +3 c_{1} -\frac {27}{2}\right ) \left (t^{3}+3 c_{1} +3 t \right )}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 523

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

Problem 524

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

Problem 525

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

Problem 526

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

Problem 527

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

Problem 528

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

Problem 529

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

Problem 530

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

Problem 531

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

Problem 532

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

Problem 533

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

Problem 534

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

Problem 535

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

Problem 536

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

Problem 537

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

Problem 538

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

Problem 539

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

Problem 540

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

Problem 541

ODE	\[ \boxed {y^{\prime }-y^{2} \left (4-y^{2}\right )=0} \]
program solution	\[ \int _{}^{y}-\frac {1}{\textit {\_a}^{2} \left (\textit {\_a}^{2}-4\right )}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-4\right ) {\mathrm e}^{\textit {\_Z}}+16 c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+16 t \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left ({\mathrm e}^{\textit {\_Z}}-4\right )-32 c_{1} +2 \textit {\_Z} -32 t +4\right )}-2 \]

Problem 542

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

Problem 543

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

Problem 544

ODE	\[ \boxed {4 y+\left (2 x -2 y\right ) y^{\prime }=-2 x} \]
program solution	\[ \frac {\ln \left (y^{2}-3 y x -x^{2}\right )}{2}+\frac {\sqrt {13}\, \operatorname {arctanh}\left (\frac {\left (-2 y+3 x \right ) \sqrt {13}}{13 x}\right )}{13} = c_{1} \] Verified OK.
Maple solution	\[ -\frac {\ln \left (\frac {-x^{2}-3 x y \left (x \right )+y \left (x \right )^{2}}{x^{2}}\right )}{2}+\frac {\sqrt {13}\, \operatorname {arctanh}\left (\frac {\left (2 y \left (x \right )-3 x \right ) \sqrt {13}}{13 x}\right )}{13}-\ln \left (x \right )-c_{1} = 0 \]

Problem 545

ODE	\[ \boxed {-2 y x +\left (3-x^{2}+6 y^{2}\right ) y^{\prime }=-3 x^{2}-2} \]
program solution	\[ x^{3}-x^{2} y+2 y^{3}+2 x +3 y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{\frac {2}{3}}+6 x^{2}-18}{6 \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{\frac {1}{3}}}{12}+\frac {\left (x^{2}-3\right ) \left (i \sqrt {3}-1\right )}{2 \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{\frac {2}{3}}}{6}+\left (-1-i \sqrt {3}\right ) \left (x^{2}-3\right )}{2 \left (-54 x^{3}-54 c_{1} -108 x +6 \sqrt {75 x^{6}+162 c_{1} x^{3}+378 x^{4}+81 c_{1}^{2}+324 c_{1} x +162 x^{2}+162}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 546

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

Problem 547

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

Problem 548

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

Problem 549

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

Problem 550

ODE	\[ \boxed {{\mathrm e}^{x} \sin \left (y\right )+3 y-\left (3 x -{\mathrm e}^{x} \sin \left (y\right )\right ) y^{\prime }=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 551

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

Problem 552

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

Problem 553

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

Problem 554

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

Problem 555

ODE	\[ \boxed {-y+\left (-x +2 y\right ) y^{\prime }=-2 x} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}
program solution	\[ y^{2}-y x +x^{2} = 7 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{2}+\frac {\sqrt {-3 x^{2}+28}}{2} \]

Problem 556

ODE	\[ \boxed {y+\left (x -4 y\right ) y^{\prime }=-9 x^{2}+1} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ 3 x^{3}+\left (y-1\right ) x -2 y^{2} = 2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{4}-\frac {\sqrt {24 x^{3}+x^{2}-8 x -16}}{4} \]

Problem 557

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

Problem 558

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

Problem 559

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

Problem 560

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

Problem 561

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

Problem 562

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

Problem 563

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

Problem 564

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

Problem 565

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

Problem 566

ODE	\[ \boxed {\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime }=-3 x} \]
program solution	\[ y x^{3}+y^{3}+3 x^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-12 x^{3}+\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}}{6 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {x^{3} \left (i \sqrt {3}-1\right )}{\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {12 i \sqrt {3}\, x^{3}+i \sqrt {3}\, \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}+12 x^{3}-\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}}{12 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 567

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

Problem 568

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

Problem 569

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

Problem 570

ODE	\[ \boxed {y^{\prime }-\frac {2 x +y}{3-x +3 y^{2}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y^{3}+\left (-x +3\right ) y-x^{2} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (108 x^{2}+12 \sqrt {81 x^{4}-12 x^{3}+108 x^{2}-324 x +324}\right )^{\frac {2}{3}}+12 x -36}{6 \left (108 x^{2}+12 \sqrt {81 x^{4}-12 x^{3}+108 x^{2}-324 x +324}\right )^{\frac {1}{3}}} \]

Problem 571

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

Problem 572

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

Problem 573

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

Problem 574

ODE	\[ \boxed {y^{\prime }-\frac {4 x^{3}+1}{y \left (2+3 y\right )}=0} \]
program solution	\[ -x^{4}+y^{3}+y^{2}-x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {2}{3}}-2 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}+4}{6 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}+4}{12 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}-4}{12 \left (-8+108 x^{4}+108 c_{1} +108 x +12 \sqrt {81}\, \sqrt {\left (x^{4}+c_{1} +x \right ) \left (x^{4}+c_{1} +x -\frac {4}{27}\right )}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 575

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

Problem 576

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

Problem 577

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

Problem 578

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

Problem 579

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

Problem 580

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

Problem 581

ODE	\[ \boxed {y+\left (x +2 y\right ) y^{\prime }=-x} \] With initial conditions \begin {align*} [y \left (2\right ) = 3] \end {align*}
program solution	\[ \frac {x^{2}}{2}+y x +y^{2} = 17 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{2}+\frac {\sqrt {-x^{2}+68}}{2} \]

Problem 582

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

Problem 583

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

Problem 584

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

Problem 585

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

Problem 586

ODE	\[ \boxed {y^{\prime }-\frac {3 x^{2}-2 y-y^{3}}{2 x +3 x y^{2}}=0} \]
program solution	\[ -x \left (-y^{3}+x^{2}-2 y\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {12^{\frac {1}{3}} \left (x^{2} 12^{\frac {1}{3}}-\frac {{\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-9 c_{1} \right ) x^{2}\right )}^{\frac {2}{3}}}{2}\right )}{3 {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-9 c_{1} \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (2 i 2^{\frac {2}{3}} 3^{\frac {5}{6}} x^{2}-2 x^{2} 2^{\frac {2}{3}} 3^{\frac {1}{3}}+i {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-9 c_{1} \right ) x^{2}\right )}^{\frac {2}{3}} \sqrt {3}+{\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-9 c_{1} \right ) x^{2}\right )}^{\frac {2}{3}}\right )}{12 x {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-9 c_{1} \right ) x^{2}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (2 \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) x^{2} 2^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-9 c_{1} \right ) x^{2}\right )}^{\frac {2}{3}}\right )}{12 {\left (\left (9 x^{3}+\sqrt {3}\, \sqrt {27 x^{6}-54 c_{1} x^{3}+27 c_{1}^{2}+32 x^{2}}-9 c_{1} \right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}

Problem 587

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

Problem 588

ODE	\[ \boxed {\frac {-4+6 y x +2 y^{2}}{3 x^{2}+4 y x +3 y^{2}}+y^{\prime }=0} \]
program solution	\[ 3 x^{2} y+2 x y^{2}+y^{3}-4 x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {1}{3}}}{6}-\frac {10 x^{2}}{3 \left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {1}{3}}}-\frac {2 x}{3} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {1}{3}}}{12}-\frac {5 x \left (i \sqrt {3}\, x -x +\frac {2 \left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {1}{3}}}{5}\right )}{3 \left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {2}{3}} \sqrt {3}+20 i \sqrt {3}\, x^{2}-\left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {2}{3}}-8 x \left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {1}{3}}+20 x^{2}}{12 \left (152 x^{3}-108 c_{1} +432 x +12 \sqrt {216 x^{6}-228 c_{1} x^{3}+912 x^{4}+81 c_{1}^{2}-648 c_{1} x +1296 x^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 589

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}-1}{1+y^{2}}=0} \] With initial conditions \begin {align*} [y \left (-1\right ) = 1] \end {align*}
program solution	\[ -\frac {x^{3}}{3}+x +\frac {y^{3}}{3}+y = {\frac {2}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (8+4 x^{3}-12 x +4 \sqrt {x^{6}-6 x^{4}+4 x^{3}+9 x^{2}-12 x +8}\right )^{\frac {2}{3}}-4}{2 \left (8+4 x^{3}-12 x +4 \sqrt {x^{6}-6 x^{4}+4 x^{3}+9 x^{2}-12 x +8}\right )^{\frac {1}{3}}} \]

Problem 590

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

Problem 591

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

Problem 592

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

Problem 593

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

Problem 594

ODE	\[ \boxed {y^{\prime }-\frac {x}{x^{2}+y+y^{3}}=0} \]
program solution	\[ -\frac {x^{2} {\mathrm e}^{-2 y}}{2}-\frac {{\mathrm e}^{-2 y} \left (4 y^{3}+6 y^{2}+10 y+5\right )}{8} = c_{1} \] Verified OK.
Maple solution	\[ \frac {\left (-4 y \left (x \right )^{3}-4 x^{2}-6 y \left (x \right )^{2}-10 y \left (x \right )-5\right ) {\mathrm e}^{-2 y \left (x \right )}}{4}+c_{1} = 0 \]

Problem 595

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

Problem 596

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

Problem 597

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

Problem 598

ODE	\[ \boxed {y^{\prime }-\frac {-3 x^{2} y-y^{2}}{2 x^{3}+3 y x}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2] \end {align*}
program solution	\[ y^{2} x \left (x^{2}+y\right ) = -4 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (i \sqrt {3}-1\right ) {\left (-\left (x^{7}-6 \sqrt {3}\, \sqrt {x^{7}+27}+54\right ) x^{2}\right )}^{\frac {2}{3}}-x^{3} \left (i \sqrt {3}\, x^{3}+x^{3}+2 {\left (-\left (x^{7}-6 \sqrt {3}\, \sqrt {x^{7}+27}+54\right ) x^{2}\right )}^{\frac {1}{3}}\right )}{6 {\left (-\left (x^{7}-6 \sqrt {3}\, \sqrt {x^{7}+27}+54\right ) x^{2}\right )}^{\frac {1}{3}} x} \]

Problem 599

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

Problem 600

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