Problem 14201
| ODE |
\[ \boxed {-x^{\prime }=-\sin \left (y \right )^{2}} \] With initial conditions \begin {align*} [x \left (0\right ) = 0] \end {align*}
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program solution |
\[ x = \frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \] Verified OK.
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Maple solution | \[ x \left (y \right ) = \frac {y}{2}-\frac {\sin \left (2 y \right )}{4} \] |
Problem 14202
| ODE |
\[ \boxed {y^{\prime }-\frac {\sqrt {t}}{y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
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program solution |
\[ -\frac {2 t^{\frac {3}{2}}}{3}+\frac {y^{2}}{2} = 2 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {2 \sqrt {9+3 t^{\frac {3}{2}}}}{3} \] |
Problem 14203
| ODE |
\[ \boxed {y^{\prime }-\sqrt {\frac {y}{t}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
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program solution |
\[ -\frac {2 \left (\sqrt {\frac {y}{t}}\, t -y\right )}{\sqrt {y}} = -2+2 \sqrt {2} \] Verified OK.
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Maple solution | \begin{align*} y \left (t \right ) &= \frac {{\left (\left (t^{2}\right )^{\frac {1}{4}} \left (\sqrt {2}-1\right )+t \right )}^{2}}{t} \\ y \left (t \right ) &= \frac {{\left (\left (t^{2}\right )^{\frac {1}{4}} \sqrt {2}+\left (t^{2}\right )^{\frac {1}{4}}-t \right )}^{2}}{t} \\ y \left (t \right ) &= \left (-2-2 \sqrt {2}\right ) \sqrt {t}+t +2 \sqrt {2}+3 \\ y \left (t \right ) &= \left (-2+2 \sqrt {2}\right ) \sqrt {t}+t -2 \sqrt {2}+3 \\ \end{align*} |
Problem 14204
| ODE |
\[ \boxed {y^{\prime }-\frac {{\mathrm e}^{t}}{y+1}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -2] \end {align*}
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program solution |
\[ \frac {y^{2}}{2}-{\mathrm e}^{t}+y = -1 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -1-\sqrt {2 \,{\mathrm e}^{t}-1} \] |
Problem 14205
| ODE |
\[ \boxed {y^{\prime }-{\mathrm e}^{t -y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \ln \left ({\mathrm e}^{t}\right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = t \] |
Problem 14206
| ODE |
\[ \boxed {y^{\prime }-\frac {y}{\ln \left (y\right )}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = {\mathrm e}] \end {align*}
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program solution |
\[ \frac {\ln \left (y\right )^{2}}{2} = x +\frac {1}{2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{\sqrt {2 x +1}} \] |
Problem 14207
| ODE |
\[ \boxed {y^{\prime }=t \sin \left (t^{2}\right )} \] With initial conditions \begin {align*} \left [y \left (\sqrt {\pi }\right ) = 0\right ] \end {align*}
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program solution |
\[ y = -\frac {\cos \left (t^{2}\right )}{2}-\frac {1}{2} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {\cos \left (t^{2}\right )}{2}-\frac {1}{2} \] |
Problem 14208
| ODE |
\[ \boxed {y^{\prime }=\frac {1}{x^{2}+1}} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \arctan \left (x \right )+1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arctan \left (x \right )+1 \] |
Problem 14209
| ODE |
\[ \boxed {y^{\prime }-\frac {\sin \left (x \right )}{\cos \left (y\right )+1}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ \cos \left (x \right )+y+\sin \left (y\right ) = 1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {RootOf}\left (-1+\cos \left (x \right )+\textit {\_Z} +\sin \left (\textit {\_Z} \right )\right ) \] |
Problem 14210
| ODE |
\[ \boxed {y^{\prime }-\frac {y+3}{3 x +1}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = 4 \left (3 x +1\right )^{\frac {1}{3}}-3 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 4 \left (3 x +1\right )^{\frac {1}{3}}-3 \] |
Problem 14211
| ODE |
\[ \boxed {y^{\prime }-{\mathrm e}^{x -y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \ln \left ({\mathrm e}^{x}-1+{\mathrm e}\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \ln \left ({\mathrm e}^{x}-1+{\mathrm e}\right ) \] |
Problem 14212
| ODE |
\[ \boxed {y^{\prime }-{\mathrm e}^{-y+2 x}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = -\ln \left (2\right )+\ln \left ({\mathrm e}^{2 x}+2 \,{\mathrm e}-1\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\ln \left (2\right )+\ln \left ({\mathrm e}^{2 x}-1+2 \,{\mathrm e}\right ) \] |
Problem 14213
| ODE |
\[ \boxed {y^{\prime }-\frac {3 y+1}{3+x}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {4}{81} x^{3}+\frac {4}{9} x^{2}+\frac {4}{3} x +1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {1}{3}+\frac {4 \left (x +3\right )^{3}}{81} \] |
Problem 14214
| ODE |
\[ \boxed {y^{\prime }-y \cos \left (t \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = {\mathrm e}^{\sin \left (t \right )} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{\sin \left (t \right )} \] |
Problem 14215
| ODE |
\[ \boxed {y^{\prime }-y^{2} \cos \left (t \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = -\frac {1}{-1+\sin \left (t \right )} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {1}{-1+\sin \left (t \right )} \] |
Problem 14216
| ODE |
\[ \boxed {y^{\prime }-\sqrt {y}\, \cos \left (t \right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = \frac {\sin \left (t \right )^{2}}{4}+\sin \left (t \right )+1 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {\left (\sin \left (t \right )+2\right )^{2}}{4} \] |
Problem 14217
| ODE |
\[ \boxed {y^{\prime }+f \left (t \right ) y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = 0 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 0 \] |
Problem 14218
| ODE |
\[ \boxed {y^{\prime }+\frac {y-2}{x -2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ y = \frac {2 x}{x -2} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2 x}{x -2} \] |
Problem 14219
| ODE |
\[ \boxed {y^{\prime }-\frac {x +y+3}{3 x +3 y+1}=0} \] |
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program solution |
\[ y = -\frac {2 \operatorname {LambertW}\left (-\frac {3 \,{\mathrm e}^{-2 x -\frac {3}{2}-2 c_{1}}}{2}\right )}{3}-x -1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {2 \operatorname {LambertW}\left (-\frac {3 \,{\mathrm e}^{-2 x -\frac {3}{2}+2 c_{1}}}{2}\right )}{3}-x -1 \] |
Problem 14220
| ODE |
\[ \boxed {y^{\prime }-\frac {x -y+2}{2 x -2 y-1}=0} \] |
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program solution |
\[ y = -\frac {5 \operatorname {LambertW}\left (-\frac {2 \,{\mathrm e}^{\frac {x}{5}-\frac {c_{1}}{5}-\frac {6}{5}}}{5}\right )}{2}+x -3 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x -\frac {5 \operatorname {LambertW}\left (-\frac {2 c_{1} {\mathrm e}^{\frac {x}{5}-\frac {6}{5}}}{5}\right )}{2}-3 \] |
Problem 14221
| ODE |
\[ \boxed {y^{\prime }-\left (x +y-4\right )^{2}=0} \] |
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program solution |
\[ y = \frac {\left (-1+\left (4-x \right ) c_{3} \right ) \cos \left (x \right )-\sin \left (x \right ) \left (-c_{3} +x -4\right )}{c_{3} \cos \left (x \right )+\sin \left (x \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -x +4-\tan \left (c_{1} -x \right ) \] |
Problem 14222
| ODE |
\[ \boxed {y^{\prime }-\left (3 y+1\right )^{4}=0} \] |
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program solution |
\[ y = \frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{9 x +9 c_{1}}-\frac {1}{3} \] Verified OK. \[ y = -\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}-\frac {1}{3} \] Verified OK. \[ y = -\frac {\left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 \left (x +c_{1} \right )}+\frac {i \sqrt {3}\, \left (-3 \left (x +c_{1} \right )^{2}\right )^{\frac {1}{3}}}{18 x +18 c_{1}}-\frac {1}{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (-\left (c_{1} +x \right )^{2}\right )^{\frac {1}{3}}-3 c_{1} -3 x}{9 c_{1} +9 x} \\ y \left (x \right ) &= \frac {\left (-i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (-\left (c_{1} +x \right )^{2}\right )^{\frac {1}{3}}-6 x -6 c_{1}}{18 c_{1} +18 x} \\ y \left (x \right ) &= \frac {\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (-\left (c_{1} +x \right )^{2}\right )^{\frac {1}{3}}-6 x -6 c_{1}}{18 c_{1} +18 x} \\ \end{align*} |
Problem 14223
| ODE |
\[ \boxed {y^{\prime }-3 y=0} \] |
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program solution |
\[ y = {\mathrm e}^{3 t} c_{1}^{3} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = c_{1} {\mathrm e}^{3 t} \] |
Problem 14224
| ODE |
\[ \boxed {y^{\prime }+y=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-t}}{c_{1}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-t} c_{1} \] |
Problem 14225
| ODE |
\[ \boxed {y^{\prime }-y^{2}+y=0} \] |
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program solution |
\[ y = -\frac {1}{-1+c_{1} {\mathrm e}^{t}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {1}{1+{\mathrm e}^{t} c_{1}} \] |
Problem 14226
| ODE |
\[ \boxed {y^{\prime }-16 y+8 y^{2}=0} \] |
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program solution |
\[ y = \frac {2 \,{\mathrm e}^{16 t} c_{1}^{16}}{-1+{\mathrm e}^{16 t} c_{1}^{16}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {2}{1+2 \,{\mathrm e}^{-16 t} c_{1}} \] |
Problem 14227
| ODE |
\[ \boxed {y^{\prime }-4 y+y^{2}=12} \] |
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program solution |
\[ y = \frac {6 \,{\mathrm e}^{8 t} c_{1}^{8}+2}{{\mathrm e}^{8 t} c_{1}^{8}-1} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {6 c_{1} {\mathrm e}^{8 t}+2}{c_{1} {\mathrm e}^{8 t}-1} \] |
Problem 14228
| ODE |
\[ \boxed {y^{\prime }-f \left (t \right ) y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ y = {\mathrm e}^{\int _{1}^{t}f \left (\textit {\_a} \right )d \textit {\_a}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{\int _{1}^{t}f \left (\textit {\_z1} \right )d \textit {\_z1}} \] |
Problem 14229
| ODE |
\[ \boxed {y^{\prime }-y=10} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{t}-10 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -10+{\mathrm e}^{t} c_{1} \] |
Problem 14230
| ODE |
\[ \boxed {y^{\prime }-y=2 \,{\mathrm e}^{-t}} \] |
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program solution |
\[ y = -\left ({\mathrm e}^{-2 t}-c_{1} \right ) {\mathrm e}^{t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -{\mathrm e}^{-t}+{\mathrm e}^{t} c_{1} \] |
Problem 14231
| ODE |
\[ \boxed {y^{\prime }-y=2 \cos \left (t \right )} \] |
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program solution |
\[ y = -{\mathrm e}^{t} \left ({\mathrm e}^{-t} \cos \left (t \right )-{\mathrm e}^{-t} \sin \left (t \right )-c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \sin \left (t \right )-\cos \left (t \right )+{\mathrm e}^{t} c_{1} \] |
Problem 14232
| ODE |
\[ \boxed {y^{\prime }-y=t^{2}-2 t} \] |
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program solution |
\[ y = -\left (t^{2} {\mathrm e}^{-t}-c_{1} \right ) {\mathrm e}^{t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -t^{2}+{\mathrm e}^{t} c_{1} \] |
Problem 14233
| ODE |
\[ \boxed {y^{\prime }-y=4 t \,{\mathrm e}^{-t}} \] |
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program solution |
\[ y = -\left (2 t \,{\mathrm e}^{-2 t}+{\mathrm e}^{-2 t}-c_{1} \right ) {\mathrm e}^{t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left (c_{1} {\mathrm e}^{2 t}-2 t -1\right ) {\mathrm e}^{-t} \] |
Problem 14234
| ODE |
\[ \boxed {t y^{\prime }+y=t^{2}} \] |
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program solution |
\[ y = \frac {t^{3}+3 c_{1}}{3 t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {t^{3}+3 c_{1}}{3 t} \] |
Problem 14235
| ODE |
\[ \boxed {t y^{\prime }+y=t} \] |
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program solution |
\[ y = \frac {t^{2}+2 c_{1}}{2 t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \frac {t}{2}+\frac {c_{1}}{t} \] |
Problem 14236
| ODE |
\[ \boxed {y^{\prime } x +y=x \,{\mathrm e}^{x}} \] |
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program solution |
\[ y = \frac {x \,{\mathrm e}^{x}-{\mathrm e}^{x}+c_{1}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (-1+x \right ) {\mathrm e}^{x}+c_{1}}{x} \] |
Problem 14237
| ODE |
\[ \boxed {y^{\prime } x +y={\mathrm e}^{-x}} \] |
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program solution |
\[ y = -\frac {{\mathrm e}^{-x}-c_{1}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-{\mathrm e}^{-x}+c_{1}}{x} \] |
Problem 14238
| ODE |
\[ \boxed {y^{\prime }-\frac {2 t y}{t^{2}+1}=2} \] |
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program solution |
\[ \frac {y}{t^{2}+1}-2 \arctan \left (t \right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left (2 \arctan \left (t \right )+c_{1} \right ) \left (t^{2}+1\right ) \] |
Problem 14239
| ODE |
\[ \boxed {y^{\prime }-\frac {4 t y}{4 t^{2}+1}=4 t} \] |
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program solution |
\[ y = \frac {c_{1} \sqrt {4 t^{2}+1}}{2}+4 t^{2}+1 \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 4 t^{2}+1+\sqrt {4 t^{2}+1}\, c_{1} \] |
Problem 14240
| ODE |
\[ \boxed {y^{\prime }-\frac {x y}{x^{2}-1}=2 x} \] |
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program solution |
\[ y = c_{1} \sqrt {x^{2}-1}+2 x^{2}-2 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 x^{2}-2+\sqrt {-1+x}\, \sqrt {1+x}\, c_{1} \] |
Problem 14241
| ODE |
\[ \boxed {y^{\prime }+y \cot \left (t \right )=\cos \left (t \right )} \] |
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program solution |
\[ y = \frac {\sin \left (t \right )^{2}+2 c_{1}}{2 \sin \left (t \right )} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = -\frac {\csc \left (t \right ) \left (2 \cos \left (t \right )^{2}-4 c_{1} -1\right )}{4} \] |
Problem 14242
| ODE |
\[ \boxed {y^{\prime }-\frac {3 t y}{t^{2}-4}=t} \] |
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program solution |
\[ y = -t^{2}+4+c_{1} \left (t^{2}-4\right )^{\frac {3}{2}} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = \left (t -2\right ) \left (t +2\right ) \left (\sqrt {t^{2}-4}\, c_{1} -1\right ) \] |
Problem 14243
| ODE |
\[ \boxed {y^{\prime }-\frac {4 t y}{4 t^{2}-9}=t} \] |
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program solution |
\[ y = t^{2}-\frac {9}{4}+c_{1} \sqrt {4 t^{2}-9} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = t^{2}-\frac {9}{4}+\sqrt {4 t^{2}-9}\, c_{1} \] |
Problem 14244
| ODE |
\[ \boxed {y^{\prime }-\frac {9 x y}{9 x^{2}+49}=x} \] |
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program solution |
\[ y = \frac {c_{1} \sqrt {9 x^{2}+49}}{3}+x^{2}+\frac {49}{9} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{2}+\frac {49}{9}+\sqrt {9 x^{2}+49}\, c_{1} \] |
Problem 14245
| ODE |
\[ \boxed {y^{\prime }+2 y \cot \left (x \right )=\cos \left (x \right )} \] |
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program solution |
\[ y = \frac {\sin \left (x \right )^{3}+3 c_{1}}{3 \sin \left (x \right )^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sin \left (x \right )}{3}+\csc \left (x \right )^{2} c_{1} \] |
Problem 14246
| ODE |
\[ \boxed {y^{\prime }+y x=x^{3}} \] |
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program solution |
\[ y = \left (x^{2} {\mathrm e}^{\frac {x^{2}}{2}}-2 \,{\mathrm e}^{\frac {x^{2}}{2}}+c_{1} \right ) {\mathrm e}^{-\frac {x^{2}}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{2}-2+{\mathrm e}^{-\frac {x^{2}}{2}} c_{1} \] |
Problem 14247
| ODE |
\[ \boxed {y^{\prime }-y x=x} \] |
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program solution |
\[ y = {\mathrm e}^{\frac {x^{2}}{2}+c_{1}}-1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -1+c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \] |
Problem 14248
| ODE |
\[ \boxed {y^{\prime }-\frac {1}{y^{2}+x}=0} \] |
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program solution |
\[ -{\mathrm e}^{-y} x -\left (y^{2}+2 y+2\right ) {\mathrm e}^{-y} = c_{1} \] Verified OK.
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Maple solution | \[ x +y \left (x \right )^{2}+2 y \left (x \right )+2-{\mathrm e}^{y \left (x \right )} c_{1} = 0 \] |
Problem 14249
| ODE |
\[ \boxed {y^{\prime }-y=x} \] |
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program solution |
\[ y = -\left (x \,{\mathrm e}^{-x}+{\mathrm e}^{-x}-c_{1} \right ) {\mathrm e}^{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -x -1+c_{1} {\mathrm e}^{x} \] |
Problem 14250
| ODE |
\[ \boxed {y-\left (x +3 y^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ -3 y+\frac {x}{y} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {c_{1}}{6}-\frac {\sqrt {c_{1}^{2}+12 x}}{6} \\ y \left (x \right ) &= -\frac {c_{1}}{6}+\frac {\sqrt {c_{1}^{2}+12 x}}{6} \\ \end{align*} |
Problem 14251
| ODE |
\[ \boxed {x^{\prime }-\frac {3 x t^{2}}{-t^{3}+1}=0} \] |
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program solution |
\[ x = \frac {{\mathrm e}^{-3 c_{1}}}{t^{3}-1} \] Verified OK.
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Maple solution | \[ x \left (t \right ) = \frac {c_{1}}{t^{3}-1} \] |
Problem 14252
| ODE |
\[ \boxed {p^{\prime }-\frac {p}{t}=t^{3}} \] |
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program solution |
\[ p = \frac {t \left (t^{3}+3 c_{1} \right )}{3} \] Verified OK.
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Maple solution | \[ p \left (t \right ) = \frac {\left (t^{3}+3 c_{1} \right ) t}{3} \] |
Problem 14253
| ODE |
\[ \boxed {v^{\prime }+v={\mathrm e}^{-s}} \] |
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program solution |
\[ v = {\mathrm e}^{-s} \left (s +c_{1} \right ) \] Verified OK.
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Maple solution | \[ v \left (s \right ) = \left (s +c_{1} \right ) {\mathrm e}^{-s} \] |
Problem 14254
| ODE |
\[ \boxed {y^{\prime }-y=4 \,{\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
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program solution |
\[ y = 4 \,{\mathrm e}^{t} t +4 \,{\mathrm e}^{t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = 4 \left (t +1\right ) {\mathrm e}^{t} \] |
Problem 14255
| ODE |
\[ \boxed {y^{\prime }+y={\mathrm e}^{-t}} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
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program solution |
\[ y = t \,{\mathrm e}^{-t}-{\mathrm e}^{-t} \] Verified OK.
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Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-t} \left (t -1\right ) \] |
Problem 14256
| ODE |
\[ \boxed {y^{\prime }+3 t^{2} y={\mathrm e}^{-t^{3}}} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
|
|
program solution |
\[ y = {\mathrm e}^{-t^{3}} t +2 \,{\mathrm e}^{-t^{3}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left (t +2\right ) {\mathrm e}^{-t^{3}} \] |
Problem 14257
| ODE |
\[ \boxed {y^{\prime }+2 y t=2 t} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
|
|
program solution |
\[ y = 1-2 \,{\mathrm e}^{-t^{2}} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 1-2 \,{\mathrm e}^{-t^{2}} \] |
Problem 14258
| ODE |
\[ \boxed {t y^{\prime }+y=\cos \left (t \right )} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{2}\right ) = \frac {4}{\pi }\right ] \end {align*}
|
|
program solution |
\[ y = \frac {\sin \left (t \right )+1}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {1+\sin \left (t \right )}{t} \] |
Problem 14259
| ODE |
\[ \boxed {t y^{\prime }+y=2 \,{\mathrm e}^{t} t} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
|
|
program solution |
\[ y = \frac {2 \,{\mathrm e}^{t} t -2 \,{\mathrm e}^{t}-1}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {-1+\left (2 t -2\right ) {\mathrm e}^{t}}{t} \] |
Problem 14260
| ODE |
\[ \boxed {\left ({\mathrm e}^{t}+1\right ) y^{\prime }+y \,{\mathrm e}^{t}=t} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
|
|
program solution |
\[ y = \frac {t^{2}-4}{2 \,{\mathrm e}^{t}+2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {t^{2}-4}{2+2 \,{\mathrm e}^{t}} \] |
Problem 14261
| ODE |
\[ \boxed {\left (t^{2}+4\right ) y^{\prime }+2 y t=2 t} \] With initial conditions \begin {align*} [y \left (0\right ) = -4] \end {align*}
|
|
program solution |
\[ y = \frac {t^{2}-16}{t^{2}+4} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {t^{2}-16}{t^{2}+4} \] |
Problem 14262
| ODE |
\[ \boxed {x^{\prime }-x=1+t} \] With initial conditions \begin {align*} [x \left (0\right ) = 2] \end {align*}
|
|
program solution |
\[ x = -2+4 \,{\mathrm e}^{t}-t \] Verified OK.
|
|
Maple solution | \[ x \left (t \right ) = -t -2+4 \,{\mathrm e}^{t} \] |
Problem 14263
| ODE |
\[ \boxed {y^{\prime }-2 y={\mathrm e}^{2 t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = t \,{\mathrm e}^{2 t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = {\mathrm e}^{2 t} t \] |
Problem 14264
| ODE |
\[ \boxed {y^{\prime }-\frac {y}{t}=\ln \left (t \right )} \] |
|
program solution |
\[ y = \frac {t \left (\ln \left (t \right )^{2}+2 c_{1} \right )}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left (\frac {\ln \left (t \right )^{2}}{2}+c_{1} \right ) t \] |
Problem 14265
| ODE |
\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{t}+\frac {y}{t^{2}}=\frac {1}{t}} \] |
|
program solution |
\[ y = t \left (c_{2} \ln \left (t \right )+c_{1} \right )+\frac {t \ln \left (t \right )^{2}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left (c_{2} +c_{1} \ln \left (t \right )+\frac {\ln \left (t \right )^{2}}{2}\right ) t \] |
Problem 14266
| ODE |
\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} 4 & 0\le t <2 \\ 0 & 2\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
N/A |
|
Maple solution | \[ y \left (t \right ) = \left \{\begin {array}{cc} 0 & t <0 \\ 4-4 \,{\mathrm e}^{-t} & t <2 \\ 4 \,{\mathrm e}^{2-t}-4 \,{\mathrm e}^{-t} & 2\le t \end {array}\right . \] |
Problem 14267
| ODE |
\[ \boxed {y^{\prime }+y=\left \{\begin {array}{cc} t & 0\le t <1 \\ 0 & 1\le t \end {array}\right .} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
|
|
program solution |
N/A |
|
Maple solution | \[ y \left (t \right ) = \left \{\begin {array}{cc} {\mathrm e}^{-t} & t <0 \\ 2 \,{\mathrm e}^{-t}+t -1 & t <1 \\ 2 \,{\mathrm e}^{-t} & 1\le t \end {array}\right . \] |
Problem 14268
| ODE |
\[ \boxed {y^{\prime }-y=\sin \left (2 t \right )} \] |
|
program solution |
\[ y = -\frac {{\mathrm e}^{t} \left ({\mathrm e}^{-t} \sin \left (2 t \right )+2 \,{\mathrm e}^{-t} \cos \left (2 t \right )-5 c_{1} \right )}{5} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {2 \cos \left (2 t \right )}{5}-\frac {\sin \left (2 t \right )}{5}+{\mathrm e}^{t} c_{1} \] |
Problem 14269
| ODE |
\[ \boxed {y^{\prime }+y=5 \,{\mathrm e}^{2 t}} \] |
|
program solution |
\[ y = \frac {\left (5 \,{\mathrm e}^{3 t}+3 c_{1} \right ) {\mathrm e}^{-t}}{3} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {5 \,{\mathrm e}^{2 t}}{3}+{\mathrm e}^{-t} c_{1} \] |
Problem 14270
| ODE |
\[ \boxed {y^{\prime }+y={\mathrm e}^{-t}} \] |
|
program solution |
\[ y = {\mathrm e}^{-t} \left (t +c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left (t +c_{1} \right ) {\mathrm e}^{-t} \] |
Problem 14271
| ODE |
\[ \boxed {y^{\prime }+y=2-{\mathrm e}^{2 t}} \] |
|
program solution |
\[ y = -\frac {\left ({\mathrm e}^{3 t}-6 \,{\mathrm e}^{t}-3 c_{1} \right ) {\mathrm e}^{-t}}{3} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 2-\frac {{\mathrm e}^{2 t}}{3}+{\mathrm e}^{-t} c_{1} \] |
Problem 14272
| ODE |
\[ \boxed {y^{\prime }-5 y=t} \] |
|
program solution |
\[ y = -\frac {\left (5 t \,{\mathrm e}^{-5 t}+{\mathrm e}^{-5 t}-25 c_{1} \right ) {\mathrm e}^{5 t}}{25} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {t}{5}-\frac {1}{25}+c_{1} {\mathrm e}^{5 t} \] |
Problem 14273
| ODE |
\[ \boxed {y^{\prime }+3 y=27 t^{2}+9} \] |
|
program solution |
\[ y = \left (9 \,{\mathrm e}^{3 t} t^{2}-6 \,{\mathrm e}^{3 t} t +5 \,{\mathrm e}^{3 t}+c_{1} \right ) {\mathrm e}^{-3 t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 9 t^{2}-6 t +5+c_{1} {\mathrm e}^{-3 t} \] |
Problem 14274
| ODE |
\[ \boxed {y^{\prime }-\frac {y}{2}=5 \cos \left (t \right )+2 \,{\mathrm e}^{t}} \] |
|
program solution |
\[ y = -{\mathrm e}^{\frac {t}{2}} \left (2 \cos \left (t \right ) {\mathrm e}^{-\frac {t}{2}}-4 \sin \left (t \right ) {\mathrm e}^{-\frac {t}{2}}-4 \,{\mathrm e}^{\frac {t}{2}}-c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 4 \sin \left (t \right )-2 \cos \left (t \right )+4 \,{\mathrm e}^{t}+{\mathrm e}^{\frac {t}{2}} c_{1} \] |
Problem 14275
| ODE |
\[ \boxed {4 y+y^{\prime }=8 \cos \left (4 t \right )} \] |
|
program solution |
\[ y = {\mathrm e}^{-4 t} \left (\cos \left (4 t \right ) {\mathrm e}^{4 t}+\sin \left (4 t \right ) {\mathrm e}^{4 t}+c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \sin \left (4 t \right )+\cos \left (4 t \right )+{\mathrm e}^{-4 t} c_{1} \] |
Problem 14276
| ODE |
\[ \boxed {y^{\prime }+10 y=2 \,{\mathrm e}^{t}} \] |
|
program solution |
\[ y = \frac {\left (2 \,{\mathrm e}^{11 t}+11 c_{1} \right ) {\mathrm e}^{-10 t}}{11} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\left (2 \,{\mathrm e}^{11 t}+11 c_{1} \right ) {\mathrm e}^{-10 t}}{11} \] |
Problem 14277
| ODE |
\[ \boxed {y^{\prime }-3 y=27 t^{2}} \] |
|
program solution |
\[ y = -\left (9 t^{2} {\mathrm e}^{-3 t}+6 \,{\mathrm e}^{-3 t} t +2 \,{\mathrm e}^{-3 t}-c_{1} \right ) {\mathrm e}^{3 t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -9 t^{2}-6 t -2+c_{1} {\mathrm e}^{3 t} \] |
Problem 14278
| ODE |
\[ \boxed {y^{\prime }-y=2 \,{\mathrm e}^{t}} \] |
|
program solution |
\[ y = {\mathrm e}^{t} \left (2 t +c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \left (2 t +c_{1} \right ) {\mathrm e}^{t} \] |
Problem 14279
| ODE |
\[ \boxed {y^{\prime }+y=4+3 \,{\mathrm e}^{t}} \] |
|
program solution |
\[ y = \frac {\left (8 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{2 t}+2 c_{1} \right ) {\mathrm e}^{-t}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = 4+\frac {3 \,{\mathrm e}^{t}}{2}+{\mathrm e}^{-t} c_{1} \] |
Problem 14280
| ODE |
\[ \boxed {y^{\prime }+y=2 \cos \left (t \right )+t} \] |
|
program solution |
\[ y = {\mathrm e}^{-t} \left (\cos \left (t \right ) {\mathrm e}^{t}+\sin \left (t \right ) {\mathrm e}^{t}+{\mathrm e}^{t} t -{\mathrm e}^{t}+c_{1} \right ) \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \sin \left (t \right )+\cos \left (t \right )+t -1+{\mathrm e}^{-t} c_{1} \] |
Problem 14281
| ODE |
\[ \boxed {y^{\prime }+\frac {y}{2}=\sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = a] \end {align*}
|
|
program solution |
\[ y = \frac {\left (2 \,{\mathrm e}^{\frac {t}{2}} \sin \left (t \right )-4 \,{\mathrm e}^{\frac {t}{2}} \cos \left (t \right )+4+5 a \right ) {\mathrm e}^{-\frac {t}{2}}}{5} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} a +\frac {2 \sin \left (t \right )}{5}+\frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{5}-\frac {4 \cos \left (t \right )}{5} \] |
Problem 14282
| ODE |
\[ \boxed {y^{\prime }-\frac {y}{2}=\sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = a] \end {align*}
|
|
program solution |
\[ y = -\frac {\left (2 \sin \left (t \right ) {\mathrm e}^{-\frac {t}{2}}+4 \cos \left (t \right ) {\mathrm e}^{-\frac {t}{2}}-4-5 a \right ) {\mathrm e}^{\frac {t}{2}}}{5} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {4 \cos \left (t \right )}{5}-\frac {2 \sin \left (t \right )}{5}+{\mathrm e}^{\frac {t}{2}} a +\frac {4 \,{\mathrm e}^{\frac {t}{2}}}{5} \] |
Problem 14283
| ODE |
\[ \boxed {t y^{\prime }+y=t \cos \left (t \right )} \] |
|
program solution |
\[ y = \frac {t \sin \left (t \right )+\cos \left (t \right )+c_{1}}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\sin \left (t \right ) t +\cos \left (t \right )+c_{1}}{t} \] |
Problem 14284
| ODE |
\[ \boxed {y^{\prime }+y=t} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = -1+{\mathrm e}^{-t}+t \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = t -1+{\mathrm e}^{-t} \] |
Problem 14285
| ODE |
\[ \boxed {y^{\prime }+y=\sin \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = \frac {\sin \left (t \right )}{2}-\frac {\cos \left (t \right )}{2}+\frac {{\mathrm e}^{-t}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = -\frac {\cos \left (t \right )}{2}+\frac {\sin \left (t \right )}{2}+\frac {{\mathrm e}^{-t}}{2} \] |
Problem 14286
| ODE |
\[ \boxed {y^{\prime }+y=\cos \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = \frac {\cos \left (t \right )}{2}+\frac {\sin \left (t \right )}{2}-\frac {{\mathrm e}^{-t}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {\cos \left (t \right )}{2}+\frac {\sin \left (t \right )}{2}-\frac {{\mathrm e}^{-t}}{2} \] |
Problem 14287
| ODE |
\[ \boxed {y^{\prime }+y={\mathrm e}^{t}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
|
|
program solution |
\[ y = -\frac {{\mathrm e}^{-t}}{2}+\frac {{\mathrm e}^{t}}{2} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{t}}{2}-\frac {{\mathrm e}^{-t}}{2} \] |
Problem 14288
| ODE |
\[ \boxed {y^{2}-\frac {y}{2 \sqrt {t}}+\left (2 y t -\sqrt {t}+1\right ) y^{\prime }=0} \] |
|
program solution |
\[ y \left (y t -\sqrt {t}\right )+y = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \frac {\sqrt {t}-1+\sqrt {t -4 c_{1} t -2 \sqrt {t}+1}}{2 t} \\ y \left (t \right ) &= \frac {\sqrt {t}-1-\sqrt {t -4 c_{1} t -2 \sqrt {t}+1}}{2 t} \\ \end{align*} |
Problem 14289
| ODE |
\[ \boxed {\frac {t}{\sqrt {t^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {t^{2}+y^{2}}}=0} \] |
|
program solution |
\[ -\frac {t^{2}}{2}-\frac {y^{2}}{2} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \sqrt {-t^{2}+c_{1}} \\ y \left (t \right ) &= -\sqrt {-t^{2}+c_{1}} \\ \end{align*} |
Problem 14290
| ODE |
\[ \boxed {y \cos \left (y t \right )+t \cos \left (y t \right ) y^{\prime }=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{-c_{1}}}{t} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= \frac {\pi }{2 t} \\ y \left (t \right ) &= -\frac {c_{1}}{t} \\ \end{align*} |
Problem 14291
| ODE |
\[ \boxed {y \sec \left (t \right )^{2}+\tan \left (t \right ) y^{\prime }=-2 t} \] |
|
program solution |
\[ y = \frac {-t^{2}+c_{1}}{\tan \left (t \right )} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \cot \left (t \right ) \left (-t^{2}+c_{1} \right ) \] |
Problem 14292
| ODE |
\[ \boxed {3 t y^{2}+y^{3} y^{\prime }=0} \] |
|
program solution |
\[ -\frac {t^{2}}{2}-\frac {y^{2}}{6} = c_{1} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= \sqrt {-3 t^{2}+c_{1}} \\ y \left (t \right ) &= -\sqrt {-3 t^{2}+c_{1}} \\ \end{align*} |
Problem 14293
| ODE |
\[ \boxed {-\sin \left (t \right ) y+\left (y^{6}+\cos \left (t \right )\right ) y^{\prime }=-t} \] |
|
program solution |
\[ \frac {t^{2}}{2}+y \cos \left (t \right )+\frac {y^{7}}{7} = c_{1} \] Verified OK.
|
|
Maple solution | \[ \frac {t^{2}}{2}+y \left (t \right ) \cos \left (t \right )+\frac {y \left (t \right )^{7}}{7}+c_{1} = 0 \] |
Problem 14294
| ODE |
\[ \boxed {\sin \left (2 t \right ) y+\left (\sqrt {y}+\cos \left (2 t \right )\right ) y^{\prime }=0} \] |
|
program solution |
\[ -\frac {\cos \left (2 t \right )}{2 y^{2}}-\frac {2}{3 y^{\frac {3}{2}}} = c_{1} \] Verified OK.
|
|
Maple solution | \[ -\frac {\cos \left (2 t \right )}{2 y \left (t \right )^{2}}-\frac {2}{3 y \left (t \right )^{\frac {3}{2}}}+c_{1} = 0 \] |
Problem 14295
| ODE |
\[ \boxed {-y-\left ({\mathrm e}^{y}-t \right ) y^{\prime }=-{\mathrm e}^{2 t}} \] |
|
program solution |
|
|
Maple solution | \[ \text {No solution found} \] |
Problem 14296
| ODE |
\[ \boxed {\ln \left (y t \right )=-\frac {t y^{\prime }}{y}} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{\frac {t +c_{1}}{t}}}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {{\mathrm e}^{\frac {t +c_{1}}{t}}}{t} \] |
Problem 14297
| ODE |
\[ \boxed {{\mathrm e}^{y t}+\frac {t \,{\mathrm e}^{y t} y^{\prime }}{y}=0} \] |
|
program solution |
\[ y = \frac {{\mathrm e}^{-c_{1}}}{t} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = \frac {c_{1}}{t} \] |
Problem 14298
| ODE |
\[ \boxed {-y^{\prime }=-3 t^{2}} \] |
|
program solution |
\[ y = t^{3}+c_{1} \] Verified OK.
|
|
Maple solution | \[ y \left (t \right ) = t^{3}+c_{1} \] |
Problem 14299
| ODE |
\[ \boxed {3 y^{2} y^{\prime }=1} \] |
|
program solution |
\[ y = \left (t +c_{1} \right )^{\frac {1}{3}} \] Verified OK. \[ y = -\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2} \] Verified OK. \[ y = -\frac {\left (t +c_{1} \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (t +c_{1} \right )^{\frac {1}{3}}}{2} \] Verified OK.
|
|
Maple solution |
\begin{align*} y \left (t \right ) &= \left (t +c_{1} \right )^{\frac {1}{3}} \\ y \left (t \right ) &= -\frac {\left (t +c_{1} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (t \right ) &= \frac {\left (t +c_{1} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*} |
Problem 14300
| ODE |
\[ \boxed {y^{2}+2 t y y^{\prime }=0} \] |
|
program solution |
\[ y = {\mathrm e}^{-\frac {\ln \left (t \right )}{2}-\frac {c_{1}}{2}} \] Verified OK.
|
|
Maple solution | \begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= \frac {\sqrt {-c_{1} t}}{t} \\ y \left (t \right ) &= -\frac {\sqrt {-c_{1} t}}{t} \\ \end{align*} |