2.17.144 Problems 14301 to 14400

Problem 14301

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

Problem 14302

ODE	\[ \boxed {y^{3}+\left (3 t y^{2}+4\right ) y^{\prime }=-2 t} \]
program solution	\[ t \left (y^{3}+t \right )+4 y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= -\frac {2 \,12^{\frac {1}{3}} \left (12^{\frac {1}{3}} t -\frac {\left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}}}{4}\right )}{3 \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {1}{3}} t} \\ y \left (t \right ) &= -\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (4 i 2^{\frac {2}{3}} 3^{\frac {5}{6}} t +i \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}} \sqrt {3}-4 \,3^{\frac {1}{3}} 2^{\frac {2}{3}} t +\left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}}\right )}{12 t \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {1}{3}}} \\ y \left (t \right ) &= \frac {\left (4 \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} t +\left (i \sqrt {3}-1\right ) \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 \left (-9 t^{2} \left (t^{2}-\frac {\sqrt {3}\, \sqrt {\frac {27 t^{5}+54 c_{1} t^{3}+27 c_{1}^{2} t +256}{t}}}{9}+c_{1} \right )\right )^{\frac {1}{3}} t} \\ \end{align*}

Problem 14303

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

Problem 14304

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

Problem 14305

ODE	\[ \boxed {2 y^{3} t +\left (1+3 y^{2} t^{2}\right ) y^{\prime }=0} \]
program solution	\[ t^{2} y^{3}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {12^{\frac {1}{3}} \left (-c_{1}^{2} 12^{\frac {1}{3}}+{\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {2}{3}}\right )}{6 c_{1} t {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {1}{3}}} \\ y \left (t \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) c_{1}^{2} 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1} t} \\ y \left (t \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) c_{1}^{2} 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}} 3^{\frac {1}{3}}}{12 {\left (\left (\sqrt {12 c_{1}^{2}+81 t^{2}}+9 t \right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1} t} \\ \end{align*}

Problem 14306

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

Problem 14307

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

Problem 14308

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

Problem 14309

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

Problem 14310

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

Problem 14311

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

Problem 14312

ODE	\[ \boxed {-y^{2} \cos \left (y t \right )+\left (t \cos \left (y t \right ) y+\sin \left (y t \right )\right ) y^{\prime }=-1} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 14313

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

Problem 14314

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

Problem 14315

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

Problem 14316

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

Problem 14317

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

Problem 14318

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

Problem 14319

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

Problem 14320

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

Problem 14321

ODE	\[ \boxed {-y-\left (t +2 y\right ) y^{\prime }=-1-5 t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ \frac {5 t^{2}}{2}-y t +t -y^{2} = 0 \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= -\frac {t}{2}-\frac {\sqrt {11 t^{2}+4 t}}{2} \\ y \left (t \right ) &= -\frac {t}{2}+\frac {\sqrt {11 t^{2}+4 t}}{2} \\ \end{align*}

Problem 14322

ODE	\[ \boxed {{\mathrm e}^{y}-2 y t +\left (t \,{\mathrm e}^{y}-t^{2}\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 14323

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

Problem 14324

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

Problem 14325

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

Problem 14326

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

Problem 14327

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

Problem 14328

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

Problem 14329

ODE	\[ \boxed {6 y \sin \left (6 y x \right )+\left (4 y^{3}+6 x \sin \left (6 y x \right )\right ) y^{\prime }=4 x^{3}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ -x^{4}-\cos \left (6 y x \right )+y^{4} = -1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-1296 x^{8}-1296 \cos \left (\textit {\_Z} \right ) x^{4}+1296 x^{4}+\textit {\_Z}^{4}\right )}{6 x} \]

Problem 14330

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

Problem 14331

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

Problem 14332

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

Problem 14333

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

Problem 14334

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

Problem 14335

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

Problem 14336

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

Problem 14337

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

Problem 14338

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

Problem 14339

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

Problem 14340

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

Problem 14341

ODE	\[ \boxed {2 y+\left (2 t +2 y\right ) y^{\prime }=-\frac {9 t}{5}} \]
program solution	\[ 2 y t +y^{2}+\frac {9 t^{2}}{10} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {-10 c_{1} t -\sqrt {10 c_{1}^{2} t^{2}+10}}{10 c_{1}} \\ y \left (t \right ) &= \frac {-10 c_{1} t +\sqrt {10 c_{1}^{2} t^{2}+10}}{10 c_{1}} \\ \end{align*}

Problem 14342

ODE	\[ \boxed {\frac {19 y}{10}+\left (\frac {19 t}{10}+2 y\right ) y^{\prime }=-2 t} \]
program solution	\[ y^{2}+\frac {19 y t}{10}+t^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {-19 c_{1} t -\sqrt {-39 c_{1}^{2} t^{2}+40}}{20 c_{1}} \\ y \left (t \right ) &= \frac {-19 c_{1} t +\sqrt {-39 c_{1}^{2} t^{2}+40}}{20 c_{1}} \\ \end{align*}

Problem 14343

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

Problem 14344

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

Problem 14345

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

Problem 14346

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

Problem 14347

ODE	\[ \boxed {y^{\prime }-2 y-\frac {\cos \left (t \right )}{\sqrt {y}}=0} \]
program solution	\[ \frac {\left (9 \cos \left (t \right )-3 \sin \left (t \right )+20 y^{\frac {3}{2}}\right ) {\mathrm e}^{-3 t}}{30} = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right )^{\frac {3}{2}}+\frac {9 \cos \left (t \right )}{20}-\frac {3 \sin \left (t \right )}{20}-c_{1} {\mathrm e}^{3 t} = 0 \]

Problem 14348

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

Problem 14349

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

Problem 14350

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

Problem 14351

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

Problem 14352

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

Problem 14353

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

Problem 14354

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

Problem 14355

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

Problem 14356

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

Problem 14357

ODE	\[ \boxed {\frac {\sin \left (2 t \right )}{\cos \left (2 y\right )}+\frac {\ln \left (y\right ) y^{\prime }}{\ln \left (t \right )}=0} \]
program solution	\[ \int _{}^{t}-\sin \left (2 \textit {\_a} \right ) \ln \left (\textit {\_a} \right )d \textit {\_a} -\frac {\sin \left (2 y\right ) \ln \left (y\right )}{2}-\frac {\pi }{4}+\frac {\operatorname {Si}\left (2 y\right )}{2} = c_{1} \] Verified OK. {y::positive}
Maple solution	\[ -\frac {i \pi \left (\operatorname {csgn}\left (t \right )-1\right ) \operatorname {csgn}\left (i t \right )}{8}+\frac {\pi \,\operatorname {csgn}\left (y \left (t \right )\right )}{8}+\frac {\sin \left (2 y \left (t \right )\right ) \ln \left (y \left (t \right )\right )}{4}-\frac {\cos \left (2 t \right ) \ln \left (t \right )}{4}-\frac {\operatorname {Si}\left (2 y \left (t \right )\right )}{4}+c_{1} +\frac {\operatorname {Ci}\left (2 t \right )}{4} = 0 \]

Problem 14358

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

Problem 14359

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

Problem 14360

ODE	\[ \boxed {t y^{\prime }+2 y=3 t} \]
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 14361

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

Problem 14362

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

Problem 14363

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

Problem 14364

ODE	\[ \boxed {y^{\prime }-\frac {t +4 y}{4 t +y}=0} \]
program solution	\[ -\frac {5 \ln \left (t -y\right )}{2}+\frac {3 \ln \left (y+t \right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (\operatorname {RootOf}\left (\textit {\_Z}^{10} c_{1} t^{2}-\textit {\_Z}^{6}-6 \textit {\_Z}^{4}-12 \textit {\_Z}^{2}-8\right )^{2}+1\right ) t \]

Problem 14365

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

Problem 14366

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

Problem 14367

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

Problem 14368

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

Problem 14369

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

Problem 14370

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

Problem 14371

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

Problem 14372

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

Problem 14373

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

Problem 14374

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

Problem 14375

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

Problem 14376

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

Problem 14377

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

Problem 14378

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

Problem 14379

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

Problem 14380

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

Problem 14381

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

Problem 14382

ODE	\[ \boxed {y^{3} t -\left (t^{4}+y^{4}\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (-2 y^{2}+t^{2}\right ) \sqrt {3}}{3 t^{2}}\right )}{3}+\ln \left (y\right ) = \frac {\sqrt {3}\, \pi }{18} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{\frac {\sqrt {3}\, \left (6 \operatorname {RootOf}\left (\sqrt {3}\, t^{2} {\mathrm e}^{-\frac {\sqrt {3}\, \pi }{9}}-3 \tan \left (\textit {\_Z} \right ) t^{2} {\mathrm e}^{-\frac {\sqrt {3}\, \pi }{9}}-2 \sqrt {3}\, {\mathrm e}^{\frac {2 \textit {\_Z} \sqrt {3}}{3}}\right )+\pi \right )}{18}} \]

Problem 14383

ODE	\[ \boxed {y^{4}+\left (t^{4}-y^{3} t \right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
program solution	\[ y = \frac {2 \,{\mathrm e}^{-\frac {8}{3}}}{\left (-\frac {8 \,{\mathrm e}^{-8}}{t^{3} \operatorname {LambertW}\left (-1, -\frac {8 \,{\mathrm e}^{-8}}{t^{3}}\right )}\right )^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {{\mathrm e}^{-\operatorname {RootOf}\left (6 i \pi \_Z220 +\operatorname {LambertW}\left (\_Z222 , -{\mathrm e}^{-3 \textit {\_Z}}\right )+3 \textit {\_Z} +3 \ln \left (2\right )+6 i \pi \_Z223 \right )}}{\left (-\frac {{\mathrm e}^{-3 \operatorname {RootOf}\left (6 i \pi \_Z220 +\operatorname {LambertW}\left (\_Z222 , -{\mathrm e}^{-3 \textit {\_Z}}\right )+3 \textit {\_Z} +3 \ln \left (2\right )+6 i \pi \_Z223 \right )}}{t^{3} \operatorname {LambertW}\left (\_Z222 , -\frac {{\mathrm e}^{-3 \operatorname {RootOf}\left (6 i \pi \_Z220 +\operatorname {LambertW}\left (\_Z222 , -{\mathrm e}^{-3 \textit {\_Z}}\right )+3 \textit {\_Z} +3 \ln \left (2\right )+6 i \pi \_Z223 \right )}}{t^{3}}\right )}\right )^{\frac {1}{3}}} \]

Problem 14384

ODE	\[ \boxed {-2 y+\left (4 t -3 y-6\right ) y^{\prime }=-1-t} \]
program solution	\[ \frac {5 \ln \left (t +3 y-9\right )}{8}-\frac {\ln \left (-t +y+1\right )}{8} = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (-t +3\right ) {\operatorname {RootOf}\left (-4+\left (3 c_{1} t^{4}-36 c_{1} t^{3}+162 c_{1} t^{2}-324 c_{1} t +243 c_{1} \right ) \textit {\_Z}^{20}-\textit {\_Z}^{4}\right )}^{4}}{3}-\frac {t}{3}+3 \]

Problem 14385

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

Problem 14386

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

Problem 14387

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

Problem 14388

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

Problem 14389

ODE	\[ \boxed {y \cot \left (x \right )+y^{\prime }-y^{4}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ \frac {1}{y^{3}} = \frac {\sin \left (x \right )^{3} \left (3 \csc \left (x \right ) \cot \left (x \right )-3 \ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )+2 c_{1} \right )}{2} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = 0 \]

Problem 14390

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

Problem 14391

ODE	\[ \boxed {t y^{\prime }-y-2 \left (t y^{\prime }-y\right )^{2}-y^{\prime }=1} \]
program solution	\[ y = t c_{1} -\frac {1}{4}+\frac {\sqrt {-8 c_{1} -7}}{4} \] Verified OK. \[ y = \frac {-7 t^{2}-2 t +1}{8 t} \] Verified OK. \[ y = c_{2} t -\frac {1}{4}-\frac {\sqrt {-8 c_{2} -7}}{4} \] Verified OK. \[ y = \frac {-7 t^{2}-2 t -3}{8 t} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {-7 t^{2}-2 t +1}{8 t} \\ y \left (t \right ) &= c_{1} t -\frac {1}{4}-\frac {\sqrt {-8 c_{1} -7}}{4} \\ y \left (t \right ) &= c_{1} t -\frac {1}{4}+\frac {\sqrt {-8 c_{1} -7}}{4} \\ \end{align*}

Problem 14392

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

Problem 14393

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

Problem 14394

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

Problem 14395

ODE	\[ \boxed {y+t y^{\prime }-\frac {{y^{\prime }}^{5}}{5}=0} \]
program solution	\[ y = 0 \] Verified OK. \[ t = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{5}-5 \textit {\_Z} t -5 y\right )^{\frac {9}{2}}+9 c_{1}}{9 \sqrt {\operatorname {RootOf}\left (\textit {\_Z}^{5}-5 \textit {\_Z} t -5 y\right )}} \] Verified OK.
Maple solution	\[ \left [t \left (\textit {\_T} \right ) = \frac {\textit {\_T}^{\frac {9}{2}}+9 c_{1}}{9 \sqrt {\textit {\_T}}}, y \left (\textit {\_T} \right ) = \frac {4 \textit {\_T}^{5}}{45}-\sqrt {\textit {\_T}}\, c_{1}\right ] \]

Problem 14396

ODE	\[ \boxed {y-t {y^{\prime }}^{2}-3 {y^{\prime }}^{2}+2 {y^{\prime }}^{3}=0} \]
program solution	\[ y = 0 \] Verified OK. \[ y = 1+t \] Verified OK. \[ t = \frac {\left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}{3}+\frac {\left (3+t \right )^{2}}{3 \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}-1+\frac {t}{3}+\frac {36 c_{1} \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}}{{\left (\left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}+\left (t -3\right ) \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}+\left (3+t \right )^{2}\right )}^{2}} \] Verified OK. \[ t = -1-\frac {\left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}{6}-\frac {\left (3+t \right )^{2}}{6 \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {t}{3}-\frac {i \left (-\left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}+\left (3+t \right )^{2}\right ) \sqrt {3}}{6 \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {144 c_{1} \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}}{{\left (\left (i \sqrt {3}-1\right ) \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}+\left (2 t -6\right ) \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}-\left (3+t \right )^{2} \left (1+i \sqrt {3}\right )\right )}^{2}} \] Verified OK. \[ t = -\frac {\left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}{6}-\frac {\left (3+t \right )^{2}}{6 \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}-1+\frac {t}{3}+\frac {i \left (-\left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}+\left (3+t \right )^{2}\right ) \sqrt {3}}{6 \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}}+\frac {144 c_{1} \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}}{{\left (\left (1+i \sqrt {3}\right ) \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {2}{3}}+\left (-2 t +6\right ) \left (-54 y+27+27 t +9 t^{2}+t^{3}+6 \sqrt {3}\, \sqrt {-y \left (27-27 y+27 t +9 t^{2}+t^{3}\right )}\right )^{\frac {1}{3}}-\left (i \sqrt {3}-1\right ) \left (3+t \right )^{2}\right )}^{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= -\frac {{\left (\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+\left (t +6\right ) \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}\right )}^{2} \left (\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+\left (-2 t -3\right ) \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}\right )}{108 t^{3}+648 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-5832 c_{1}} \\ y \left (t \right ) &= \frac {{\left (i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, t^{2}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}-2 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}-12 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}\right )}^{2} \left (i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}-i \sqrt {3}\, t^{2}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}+4 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+t^{2}+6 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}\right )}{864 t^{3}+5184 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-46656 c_{1}} \\ y \left (t \right ) &= \frac {\left (i \sqrt {3}\, t^{2}-i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+t^{2}-2 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}-12 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )^{2} \left (i \sqrt {3}\, t^{2}-i \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}} \sqrt {3}+t^{2}+4 t \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+6 \left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {1}{3}}+\left (t^{3}+6 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-54 c_{1} \right )^{\frac {2}{3}}\right )}{864 t^{3}+5184 \sqrt {3}\, \sqrt {-c_{1} \left (t^{3}-27 c_{1} \right )}-46656 c_{1}} \\ \end{align*}

Problem 14397

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

Problem 14398

ODE	\[ \boxed {y-t \left (2-y^{\prime }\right )-2 {y^{\prime }}^{2}=1} \]
program solution	\[ y = 3+t \] Verified OK. \[ t = \frac {t}{3}+\frac {\sqrt {t^{2}+8 y-16 t -8}}{3}+\frac {8}{3}+\frac {2 c_{1}}{\sqrt {t +\sqrt {t^{2}+8 y-16 t -8}-4}} \] Verified OK. \[ t = \frac {t}{3}-\frac {\sqrt {t^{2}+8 y-16 t -8}}{3}+\frac {8}{3}+\frac {2 c_{1}}{\sqrt {t -\sqrt {t^{2}+8 y-16 t -8}-4}} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= -\frac {\left (\frac {t -4}{\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {1}{3}}}+\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {1}{3}}-2\right ) t \left (\frac {t -4}{\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {1}{3}}}+\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {1}{3}}+2\right )}{4}+\frac {\left (\frac {\left (t -4\right )^{2}}{\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {2}{3}}}+2 t -4+\left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {2}{3}}\right )^{2}}{8}+1 \\ y \left (t \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (t^{3}-6 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+18 c_{1}^{2}-12 t^{2}+48 t -64\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {2}{3}}+12 \left (c_{1} -\frac {\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}}{3}\right ) \left (t^{2}-4 t +28\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {1}{3}}+\left (t -4\right ) \left (t^{3}-12 t^{2}-36 c_{1}^{2}+12 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+48 t -64\right ) \left (i \sqrt {3}-1\right )}{16 \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {4}{3}}} \\ y \left (t \right ) &= \frac {\left (t^{3}-6 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+18 c_{1}^{2}-12 t^{2}+48 t -64\right ) \left (i \sqrt {3}-1\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {2}{3}}-12 \left (c_{1} -\frac {\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}}{3}\right ) \left (t^{2}-4 t +28\right ) \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) \left (t -4\right ) \left (t^{3}-12 t^{2}-36 c_{1}^{2}+12 c_{1} \sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}+48 t -64\right )}{16 \left (-3 c_{1} +\sqrt {-t^{3}+9 c_{1}^{2}+12 t^{2}-48 t +64}\right )^{\frac {4}{3}}} \\ \end{align*}

Problem 14399

ODE	\[ \boxed {t^{\frac {1}{3}} y^{\frac {2}{3}}+\left (t^{\frac {2}{3}} y^{\frac {1}{3}}+y\right ) y^{\prime }=-t} \]
program solution	\[ \ln \left (t \right ) = -\frac {3 \ln \left (\left (\frac {y}{t}\right )^{\frac {4}{3}}+1\right )}{4}+c_{1} \] Verified OK.
Maple solution	\[ -\left (\int _{\textit {\_b}}^{t}\frac {y \left (t \right )^{\frac {8}{3}} \textit {\_a}^{\frac {1}{3}}-y \left (t \right )^{\frac {4}{3}} \textit {\_a}^{\frac {5}{3}}+\textit {\_a}^{3}}{\textit {\_a}^{4}+y \left (t \right )^{4}}d \textit {\_a} \right )-\frac {\left (\int _{}^{y \left (t \right )}\frac {\left (4 t^{2}+4 \textit {\_f}^{\frac {2}{3}} t^{\frac {4}{3}}+4 \textit {\_f}^{\frac {4}{3}} t^{\frac {2}{3}}+4 \textit {\_f}^{2}\right ) \left (\int _{\textit {\_b}}^{t}\frac {\textit {\_a}^{\frac {17}{3}} \textit {\_f}^{\frac {1}{3}}-2 \textit {\_a}^{\frac {13}{3}} \textit {\_f}^{\frac {5}{3}}-2 \textit {\_a}^{\frac {5}{3}} \textit {\_f}^{\frac {13}{3}}+\textit {\_a}^{\frac {1}{3}} \textit {\_f}^{\frac {17}{3}}+3 \textit {\_a}^{3} \textit {\_f}^{3}}{\left (\textit {\_a}^{4}+\textit {\_f}^{4}\right )^{2}}d \textit {\_a} \right )+3 t^{\frac {2}{3}} \textit {\_f}^{\frac {1}{3}}+3 \textit {\_f}}{\textit {\_f}^{\frac {4}{3}} t^{\frac {2}{3}}+\textit {\_f}^{\frac {2}{3}} t^{\frac {4}{3}}+t^{2}+\textit {\_f}^{2}}d \textit {\_f} \right )}{3}+c_{1} = 0 \]

Problem 14400

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