2.17.130 Problems 12901 to 13000

Problem 12901

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

Problem 12902

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

Problem 12903

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

Problem 12904

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

Problem 12905

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

Problem 12906

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

Problem 12907

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

Problem 12908

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

Problem 12909

ODE	\[ \boxed {S^{\prime }-S^{3}+2 S^{2}-S=0} \] With initial conditions \begin {align*} \left [S \left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
program solution	\[ -\frac {1}{S-1}-\ln \left (S-1\right )+\ln \left (S\right ) = -i \pi +t +2 \] Verified OK.
Maple solution	\[ S \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (-i \pi \,{\mathrm e}^{\textit {\_Z}}-\ln \left ({\mathrm e}^{\textit {\_Z}}+1\right ) {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+t \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}+1\right )}+1 \]

Problem 12910

ODE	\[ \boxed {S^{\prime }-S^{3}+2 S^{2}-S=0} \] With initial conditions \begin {align*} \left [S \left (1\right ) = {\frac {1}{2}}\right ] \end {align*}
program solution	\[ -\frac {1}{S-1}-\ln \left (S-1\right )+\ln \left (S\right ) = -i \pi +t +1 \] Verified OK.
Maple solution	\[ S \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (-i \pi \,{\mathrm e}^{\textit {\_Z}}-\ln \left ({\mathrm e}^{\textit {\_Z}}+1\right ) {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+t \,{\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{\textit {\_Z}}+1\right )}+1 \]

Problem 12911

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

Problem 12912

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

Problem 12913

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

Problem 12914

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

Problem 12915

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

Problem 12916

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

Problem 12917

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

Problem 12918

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

Problem 12919

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

Problem 12920

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

Problem 12921

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

Problem 12922

ODE	\[ \boxed {\theta ^{\prime }+\frac {11 \cos \left (\theta \right )}{10}={\frac {9}{10}}} \]
program solution	\[ \theta = -2 \arctan \left (\frac {\tanh \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \] Verified OK.
Maple solution	\[ \theta \left (t \right ) = -2 \arctan \left (\frac {\tanh \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \]

Problem 12923

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

Problem 12924

ODE	\[ \boxed {\theta ^{\prime }+\frac {9 \cos \left (\theta \right )}{10}={\frac {11}{10}}} \]
program solution	\[ \theta = 2 \arctan \left (\frac {\tan \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \] Verified OK.
Maple solution	\[ \theta \left (t \right ) = 2 \arctan \left (\frac {\tan \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \]

Problem 12925

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

Problem 12926

ODE	\[ \boxed {v^{\prime }-\frac {K -v}{R C}=0} \]
program solution	\[ v = -\frac {{\mathrm e}^{-\frac {t}{R C}}}{c_{1}}+K \] Verified OK.
Maple solution	\[ v \left (t \right ) = K +c_{1} {\mathrm e}^{-\frac {t}{R C}} \]

Problem 12927

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

Problem 12928

ODE	\[ \boxed {y^{\prime }-2 y=1} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ -\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 y+1\right )}{2} = t +\frac {\ln \left (7\right )}{2}-\frac {\ln \left (2\right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {1}{2}+\frac {7 \,{\mathrm e}^{2 t}}{2} \]

Problem 12929

ODE	\[ \boxed {y^{\prime }+y^{2}=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {-2 \operatorname {AiryAi}\left (1, t\right ) \pi 3^{\frac {5}{6}}+3 \operatorname {AiryAi}\left (1, t\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}+3 \operatorname {AiryBi}\left (1, t\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {1}{6}}+2 \operatorname {AiryBi}\left (1, t\right ) \pi 3^{\frac {1}{3}}}{-2 \operatorname {AiryAi}\left (t \right ) \pi 3^{\frac {5}{6}}+3 \operatorname {AiryAi}\left (t \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}+3 \operatorname {AiryBi}\left (t \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {1}{6}}+2 \operatorname {AiryBi}\left (t \right ) \pi 3^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2 \operatorname {AiryAi}\left (1, t\right ) \pi 3^{\frac {5}{6}}-3 \operatorname {AiryAi}\left (1, t\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}-3 \operatorname {AiryBi}\left (1, t\right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}-2 \operatorname {AiryBi}\left (1, t\right ) \pi 3^{\frac {1}{3}}}{2 \operatorname {AiryAi}\left (t \right ) \pi 3^{\frac {5}{6}}-3 \operatorname {AiryAi}\left (t \right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}}-3 \operatorname {AiryBi}\left (t \right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}-2 \operatorname {AiryBi}\left (t \right ) \pi 3^{\frac {1}{3}}} \]

Problem 12930

ODE	\[ \boxed {y^{\prime }-y^{2}=-4 t} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {1}{2}}\right ] \end {align*}
program solution	\[ y = \frac {-2^{\frac {2}{3}} \operatorname {AiryAi}\left (1, 2^{\frac {2}{3}} t \right ) 3^{\frac {5}{6}} \pi -3 \,2^{\frac {2}{3}} \operatorname {AiryAi}\left (1, 2^{\frac {2}{3}} t \right ) \Gamma \left (\frac {2}{3}\right )^{2} 6^{\frac {2}{3}}+2^{\frac {2}{3}} \operatorname {AiryBi}\left (1, 2^{\frac {2}{3}} t \right ) 3^{\frac {1}{3}} \pi -6 \,2^{\frac {1}{3}} \operatorname {AiryBi}\left (1, 2^{\frac {2}{3}} t \right ) 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}}{\operatorname {AiryAi}\left (2^{\frac {2}{3}} t \right ) 3^{\frac {5}{6}} \pi +3 \operatorname {AiryBi}\left (2^{\frac {2}{3}} t \right ) 3^{\frac {1}{6}} 2^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )^{2}+3 \operatorname {AiryAi}\left (2^{\frac {2}{3}} t \right ) \Gamma \left (\frac {2}{3}\right )^{2} 6^{\frac {2}{3}}-\operatorname {AiryBi}\left (2^{\frac {2}{3}} t \right ) 3^{\frac {1}{3}} \pi } \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2^{\frac {2}{3}} \left (\left (3 \,2^{\frac {2}{3}} 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}-\pi 3^{\frac {1}{3}}\right ) \operatorname {AiryBi}\left (1, 2^{\frac {2}{3}} t \right )+\operatorname {AiryAi}\left (1, 2^{\frac {2}{3}} t \right ) \left (3 \Gamma \left (\frac {2}{3}\right )^{2} 6^{\frac {2}{3}}+3^{\frac {5}{6}} \pi \right )\right )}{\left (-3 \Gamma \left (\frac {2}{3}\right )^{2} 6^{\frac {2}{3}}-3^{\frac {5}{6}} \pi \right ) \operatorname {AiryAi}\left (2^{\frac {2}{3}} t \right )+\operatorname {AiryBi}\left (2^{\frac {2}{3}} t \right ) \left (-3 \,2^{\frac {2}{3}} 3^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )^{2}+\pi 3^{\frac {1}{3}}\right )} \]

Problem 12931

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

Problem 12932

ODE	\[ \boxed {w^{\prime }-\left (3-w\right ) \left (w+1\right )=0} \] With initial conditions \begin {align*} [w \left (0\right ) = 4] \end {align*}
program solution	\[ \frac {\ln \left (w+1\right )}{4}-\frac {\ln \left (w-3\right )}{4} = t +\frac {\ln \left (5\right )}{4} \] Verified OK.
Maple solution	\[ w \left (t \right ) = \frac {15 \,{\mathrm e}^{4 t}+1}{-1+5 \,{\mathrm e}^{4 t}} \]

Problem 12933

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

Problem 12934

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

Problem 12935

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

Problem 12936

ODE	\[ \boxed {y^{\prime }-y^{2}+y^{3}=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = {\frac {1}{5}}\right ] \end {align*}
program solution	\[ -\frac {1}{y}+\ln \left (y\right )-\ln \left (y-1\right ) = t -i \pi -5-2 \ln \left (2\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {1}{\operatorname {LambertW}\left (4 \,{\mathrm e}^{4-t}\right )+1} \]

Problem 12937

ODE	\[ \boxed {y^{\prime }-2 y^{3}=t^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {1}{2}}\right ] \end {align*}
program solution
Maple solution	\[ \text {No solution found} \]

Problem 12938

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

Problem 12939

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

Problem 12940

ODE	\[ \boxed {\theta ^{\prime }+\frac {11 \cos \left (\theta \right )}{10}={\frac {9}{10}}} \] With initial conditions \begin {align*} [\theta \left (0\right ) = 1] \end {align*}
program solution	\[ -\sqrt {10}\, \operatorname {arctanh}\left (\tan \left (\frac {\theta }{2}\right ) \sqrt {10}\right ) = t -\sqrt {10}\, \operatorname {arccoth}\left (\tan \left (\frac {1}{2}\right ) \sqrt {10}\right )+\frac {i \pi \sqrt {10}}{2} \] Verified OK.
Maple solution	\[ \theta \left (t \right ) = -2 \arctan \left (\frac {\tanh \left (-\operatorname {arctanh}\left (\tan \left (\frac {1}{2}\right ) \sqrt {10}\right )+\frac {\sqrt {10}\, t}{10}\right ) \sqrt {10}}{10}\right ) \]

Problem 12941

ODE	\[ \boxed {y^{\prime }-y \left (y-1\right ) \left (y-3\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ \frac {\ln \left (y\right )}{3}-\frac {\ln \left (y-1\right )}{2}+\frac {\ln \left (y-3\right )}{6} = t +\frac {2 \ln \left (2\right )}{3}-\frac {\ln \left (3\right )}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {48 \left (\frac {{\mathrm e}^{6 t}}{3}-\frac {9}{16}\right ) \left (27-32 \,{\mathrm e}^{6 t}+8 \sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}\right )^{\frac {2}{3}}+48 \left (\left (27-32 \,{\mathrm e}^{6 t}+8 \sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}\right )^{\frac {1}{3}}+3\right ) \left ({\mathrm e}^{6 t}-\frac {\sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}}{4}-\frac {27}{16}\right )}{\left (27-32 \,{\mathrm e}^{6 t}+8 \sqrt {16 \,{\mathrm e}^{12 t}-27 \,{\mathrm e}^{6 t}}\right )^{\frac {2}{3}} \left (16 \,{\mathrm e}^{6 t}-27\right )} \]

Problem 12942

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

Problem 12943

ODE	\[ \boxed {y^{\prime }-y \left (y-1\right ) \left (y-3\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ \frac {\ln \left (y\right )}{3}-\frac {\ln \left (y-1\right )}{2}+\frac {\ln \left (y-3\right )}{6} = t +\frac {\ln \left (2\right )}{3}+\frac {i \pi }{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (16 \,{\mathrm e}^{6 t}+9\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{\frac {2}{3}}+\left (24 \,{\mathrm e}^{6 t}+12 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+9\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{\frac {1}{3}}+48 \,{\mathrm e}^{6 t}+24 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+9}{\left (16 \,{\mathrm e}^{6 t}+3\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{\frac {2}{3}}+\left (8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+3\right ) \left (1+8 \,{\mathrm e}^{6 t}+4 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}\right )^{\frac {1}{3}}+16 \,{\mathrm e}^{6 t}+8 \sqrt {{\mathrm e}^{6 t}+4 \,{\mathrm e}^{12 t}}+3} \]

Problem 12944

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

Problem 12945

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

Problem 12946

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

Problem 12947

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

Problem 12948

ODE	\[ \boxed {y^{\prime }-\frac {1}{\left (y+2\right )^{2}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ \frac {\left (y+2\right )^{3}}{3} = t +9 \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (3 t +27\right )^{\frac {1}{3}}-2 \]

Problem 12949

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

Problem 12950

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

Problem 12951

ODE	\[ \boxed {y^{\prime }-3 y \left (y-2\right )=0} \] With initial conditions \begin {align*} [y \left (-2\right ) = -1] \end {align*}
program solution	\[ \frac {\ln \left (y-2\right )}{6}-\frac {\ln \left (y\right )}{6} = t +2+\frac {\ln \left (3\right )}{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {2}{3 \,{\mathrm e}^{6 t +12}-1} \]

Problem 12952

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

Problem 12953

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

Problem 12954

ODE	\[ \boxed {y^{\prime }-y^{2}+4 y=-12} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ \frac {\ln \left (y-6\right )}{8}-\frac {\ln \left (y+2\right )}{8} = t +\frac {\ln \left (5\right )}{8}+\frac {i \pi }{8}-\frac {\ln \left (3\right )}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {18-10 \,{\mathrm e}^{8 t}}{5 \,{\mathrm e}^{8 t}+3} \]

Problem 12955

ODE	\[ \boxed {y^{\prime }-y^{2}+4 y=-12} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ \frac {\ln \left (y-6\right )}{8}-\frac {\ln \left (y+2\right )}{8} = t -1+\frac {i \pi }{8}+\frac {\ln \left (3\right )}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {6-6 \,{\mathrm e}^{-8+8 t}}{3 \,{\mathrm e}^{-8+8 t}+1} \]

Problem 12956

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

Problem 12957

ODE	\[ \boxed {y^{\prime }-y^{2}+4 y=-12} \] With initial conditions \begin {align*} [y \left (0\right ) = 5] \end {align*}
program solution	\[ \frac {\ln \left (y-6\right )}{8}-\frac {\ln \left (y+2\right )}{8} = t +\frac {i \pi }{8}-\frac {\ln \left (7\right )}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {42-2 \,{\mathrm e}^{8 t}}{{\mathrm e}^{8 t}+7} \]

Problem 12958

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

Problem 12959

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

Problem 12960

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

Problem 12961

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

Problem 12962

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

Problem 12963

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

Problem 12964

ODE	\[ \boxed {w^{\prime }-w \cos \left (w\right )=0} \] With initial conditions \begin {align*} [w \left (3\right ) = 1] \end {align*}
program solution	\[ \int _{0}^{w}\frac {1}{\textit {\_a} \cos \left (\textit {\_a} \right )}d \textit {\_a} +\int _{0}^{1}\frac {1}{w \cos \left (w \right )}d \textit {\_a} = t -3 \] Verified OK.
Maple solution	\[ w \left (t \right ) = \operatorname {RootOf}\left (\int _{\textit {\_Z}}^{1}\frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} +t -3\right ) \]

Problem 12965

ODE	\[ \boxed {w^{\prime }-w \cos \left (w\right )=0} \] With initial conditions \begin {align*} [w \left (0\right ) = 2] \end {align*}
program solution	\[ \int _{0}^{w}\frac {1}{\textit {\_a} \cos \left (\textit {\_a} \right )}d \textit {\_a} +\int _{0}^{2}\frac {1}{w \cos \left (w \right )}d \textit {\_a} = t \] Verified OK.
Maple solution	\[ w \left (t \right ) = \operatorname {RootOf}\left (\int _{\textit {\_Z}}^{2}\frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} +t \right ) \]

Problem 12966

ODE	\[ \boxed {w^{\prime }-w \cos \left (w\right )=0} \] With initial conditions \begin {align*} [w \left (0\right ) = -1] \end {align*}
program solution	\[ \int _{0}^{w}\frac {1}{\textit {\_a} \cos \left (\textit {\_a} \right )}d \textit {\_a} +\int _{0}^{-1}\frac {1}{w \cos \left (w \right )}d \textit {\_a} = t \] Verified OK.
Maple solution	\[ w \left (t \right ) = \operatorname {RootOf}\left (\int _{\textit {\_Z}}^{-1}\frac {\sec \left (\textit {\_a} \right )}{\textit {\_a}}d \textit {\_a} +t \right ) \]

Problem 12967

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

Problem 12968

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

Problem 12969

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

Problem 12970

ODE	\[ \boxed {w^{\prime }-3 w^{3}+12 w^{2}=0} \]
program solution	\[ \int _{}^{w}\frac {1}{3 \textit {\_a}^{3}-12 \textit {\_a}^{2}}d \textit {\_a} = t +c_{1} \] Verified OK.
Maple solution	\[ w \left (t \right ) = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}+4\right ) {\mathrm e}^{\textit {\_Z}}+48 c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+48 t \,{\mathrm e}^{\textit {\_Z}}+4 \ln \left ({\mathrm e}^{\textit {\_Z}}+4\right )+192 c_{1} -4 \textit {\_Z} +192 t -4\right )}+4 \]

Problem 12971

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

Problem 12972

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

Problem 12973

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

Problem 12974

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

Problem 12975

ODE	\[ \boxed {y^{\prime }-y^{2}+4 y=2} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
program solution	\[ -\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (y-2\right ) \sqrt {2}}{2}\right )}{2} = t +\frac {\operatorname {arccoth}\left (\frac {3 \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {i \pi \sqrt {2}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\sqrt {2}\, \tanh \left (\operatorname {arctanh}\left (\frac {3 \sqrt {2}}{2}\right )+\sqrt {2}\, t \right )+2 \]

Problem 12976

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

Problem 12977

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

Problem 12978

ODE	\[ \boxed {y^{\prime }-y^{2}+4 y=2} \] With initial conditions \begin {align*} [y \left (0\right ) = -4] \end {align*}
program solution	\[ -\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (y-2\right ) \sqrt {2}}{2}\right )}{2} = t +\frac {\sqrt {2}\, \operatorname {arccoth}\left (3 \sqrt {2}\right )}{2}-\frac {i \pi \sqrt {2}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\sqrt {2}\, \tanh \left (\operatorname {arctanh}\left (3 \sqrt {2}\right )+\sqrt {2}\, t \right )+2 \]

Problem 12979

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

Problem 12980

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

Problem 12981

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

Problem 12982

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

Problem 12983

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

Problem 12984

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

Problem 12985

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

Problem 12986

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

Problem 12987

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

Problem 12988

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

Problem 12989

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

Problem 12990

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

Problem 12991

ODE	\[ \boxed {y^{\prime }+3 y=4 \cos \left (2 t \right )} \]
program solution	\[ y = \frac {{\mathrm e}^{-3 t} \left (12 \,{\mathrm e}^{3 t} \cos \left (2 t \right )+8 \,{\mathrm e}^{3 t} \sin \left (2 t \right )+13 c_{1} \right )}{13} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {12 \cos \left (2 t \right )}{13}+\frac {8 \sin \left (2 t \right )}{13}+c_{1} {\mathrm e}^{-3 t} \]

Problem 12992

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

Problem 12993

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

Problem 12994

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

Problem 12995

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

Problem 12996

ODE	\[ \boxed {y^{\prime }-2 y=3 \,{\mathrm e}^{-2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 10] \end {align*}
program solution	\[ y = -\frac {3 \,{\mathrm e}^{-2 t}}{4}+\frac {43 \,{\mathrm e}^{2 t}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {43 \,{\mathrm e}^{2 t}}{4}-\frac {3 \,{\mathrm e}^{-2 t}}{4} \]

Problem 12997

ODE	\[ \boxed {y^{\prime }+y=\cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 5] \end {align*}
program solution	\[ y = \frac {2 \sin \left (2 t \right )}{5}+\frac {\cos \left (2 t \right )}{5}+\frac {24 \,{\mathrm e}^{-t}}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\cos \left (2 t \right )}{5}+\frac {2 \sin \left (2 t \right )}{5}+\frac {24 \,{\mathrm e}^{-t}}{5} \]

Problem 12998

ODE	\[ \boxed {y^{\prime }+3 y=\cos \left (2 t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
program solution	\[ y = \frac {2 \sin \left (2 t \right )}{13}+\frac {3 \cos \left (2 t \right )}{13}-\frac {16 \,{\mathrm e}^{-3 t}}{13} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {3 \cos \left (2 t \right )}{13}+\frac {2 \sin \left (2 t \right )}{13}-\frac {16 \,{\mathrm e}^{-3 t}}{13} \]

Problem 12999

ODE	\[ \boxed {y^{\prime }-2 y=7 \,{\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ y = 7 t \,{\mathrm e}^{2 t}+3 \,{\mathrm e}^{2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (7 t +3\right ) {\mathrm e}^{2 t} \]

Problem 13000

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