2.17.145 Problems 14401 to 14500

Problem 14401

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

Problem 14402

ODE	\[ \boxed {y^{\prime }-\frac {2 t^{5}}{5 y^{2}}=0} \]
program solution	\[ -\frac {t^{6}}{6}+\frac {5 y^{3}}{6} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {\left (25 t^{6}+125 c_{1} \right )^{\frac {1}{3}}}{5} \\ y \left (t \right ) &= -\frac {\left (25 t^{6}+125 c_{1} \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{10} \\ y \left (t \right ) &= \frac {\left (25 t^{6}+125 c_{1} \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{10} \\ \end{align*}

Problem 14403

ODE	\[ \boxed {-8 y^{\prime } \sin \left (y\right )=-\cos \left (4 x \right )} \]
program solution	\[ -\frac {\sin \left (4 x \right )}{4}-8 \cos \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\pi }{2}+\arcsin \left (\frac {\sin \left (4 x \right )}{32}+\frac {c_{1}}{8}\right ) \]

Problem 14404

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 14405

ODE	\[ \boxed {y^{\prime }-\frac {{\mathrm e}^{8 y}}{t}=0} \]
program solution	\[ y = -\frac {\ln \left (-8 \ln \left (t \right )-8 c_{1} \right )}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {3 \ln \left (2\right )}{8}-\frac {\ln \left (-\ln \left (t \right )-c_{1} \right )}{8} \]

Problem 14406

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

Problem 14407

ODE	\[ \boxed {-\left (2 y^{4}-6 y^{9}\right ) y^{\prime }=\frac {1}{x^{5}}-\frac {1}{x^{3}}} \]
program solution	\[ \frac {2 x^{2}-1}{4 x^{4}}-\frac {\left (3 y^{5}-1\right )^{2}}{15} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {6^{\frac {4}{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{6 x} \\ y \left (x \right ) &= \frac {6^{\frac {4}{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{6 x} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {10-2 \sqrt {5}}+\sqrt {5}+1\right ) 6^{\frac {4}{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {10-2 \sqrt {5}}-\sqrt {5}-1\right ) 6^{\frac {4}{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {10+2 \sqrt {5}}-\sqrt {5}+1\right ) 6^{\frac {4}{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {10+2 \sqrt {5}}+\sqrt {5}-1\right ) 6^{\frac {4}{5}} \left (2 x^{5}+x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {10-2 \sqrt {5}}+\sqrt {5}+1\right ) 6^{\frac {4}{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {10-2 \sqrt {5}}-\sqrt {5}-1\right ) 6^{\frac {4}{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {10+2 \sqrt {5}}-\sqrt {5}+1\right ) 6^{\frac {4}{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {10+2 \sqrt {5}}+\sqrt {5}-1\right ) 6^{\frac {4}{5}} \left (2 x^{5}-x^{3} \sqrt {-60 c_{1} x^{4}+30 x^{2}-15}\right )^{\frac {1}{5}}}{24 x} \\ \end{align*}

Problem 14408

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

Problem 14409

ODE	\[ \boxed {y^{\prime }-\frac {\left (4-7 x \right ) \left (2 y-3\right )}{\left (x -1\right ) \left (2 x -5\right )}=0} \]
program solution	\[ y = \frac {1536 x^{9}-34560 x^{8}+345600 x^{7}-2016000 x^{6}+7560000 x^{5}-18900000 x^{4}+x^{2} {\mathrm e}^{-2 c_{1}}[phone] x^{3}-2 \,{\mathrm e}^{-2 c_{1}} x -33750000 x^{2}+{\mathrm e}^{-2 c_{1}}[phone] x -5859375}{2 \left (2 x -5\right )^{9}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1536 x^{9}-34560 x^{8}+345600 x^{7}-2016000 x^{6}+7560000 x^{5}-18900000 x^{4}[phone] x^{3}+\left (2 c_{1} -28303968\right ) x^{2}+\left (-4 c_{1} [phone]\right ) x +2 c_{1} -413343}{2 \left (2 x -5\right )^{9}} \]

Problem 14410

ODE	\[ \boxed {y^{\prime }+3 y=-10 \sin \left (t \right )} \]
program solution	\[ y = -{\mathrm e}^{-3 t} \left (3 \,{\mathrm e}^{3 t} \sin \left (t \right )-{\mathrm e}^{3 t} \cos \left (t \right )-c_{1} \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cos \left (t \right )-3 \sin \left (t \right )+c_{1} {\mathrm e}^{-3 t} \]

Problem 14411

ODE	\[ \boxed {\left (-4 y+t \right ) y^{\prime }=-3 t} \]
program solution	\[ \frac {3 \ln \left (-t +y\right )}{7}+\frac {4 \ln \left (3 t +4 y\right )}{7} = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {t \left (\operatorname {RootOf}\left (\textit {\_Z}^{49} c_{1} t^{7}-21 \textit {\_Z}^{42} c_{1} t^{7}+147 \textit {\_Z}^{35} c_{1} t^{7}-343 \textit {\_Z}^{28} c_{1} t^{7}-64\right )^{7}-3\right )}{4} \]

Problem 14412

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

Problem 14413

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

Problem 14414

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

Problem 14415

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

Problem 14416

ODE	\[ \boxed {x^{\prime }-\frac {5 t x}{x^{2}+t^{2}}=0} \]
program solution	\[ -\frac {5 t^{2}}{2 x^{\frac {2}{5}}}+\frac {5 x^{\frac {8}{5}}}{8} = c_{1} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {4 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{72}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{64}+\operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{56}+1024 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{32} t^{8} c_{1} -1280 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{24} t^{8} c_{1} +640 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{16} t^{8} c_{1} -160 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{8} t^{8} c_{1} +18 c_{1} t^{8}}{c_{1} t^{7}} \]

Problem 14417

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

Problem 14418

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

Problem 14419

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

Problem 14420

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

Problem 14421

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

Problem 14422

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

Problem 14423

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

Problem 14424

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

Problem 14425

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

Problem 14426

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

Problem 14427

ODE	\[ \boxed {y-t y^{\prime }-3 {y^{\prime }}^{4}=0} \]
program solution	\[ y = 3 c_{1}^{4}+c_{1} t \] Verified OK. \[ y = -\frac {18^{\frac {1}{3}} \left (-t \right )^{\frac {4}{3}}}{8} \] Verified OK. \[ y = \frac {2^{\frac {1}{3}} \left (-t \right )^{\frac {4}{3}} \left (3 i 3^{\frac {1}{6}}+3^{\frac {2}{3}}\right )}{16} \] Verified OK. \[ y = -\frac {3 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {1}{3}} \left (-t \right )^{\frac {4}{3}}}{16} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= -\frac {18^{\frac {1}{3}} \left (-t \right )^{\frac {4}{3}}}{8} \\ y \left (t \right ) &= \frac {18^{\frac {1}{3}} \left (-t \right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}{16} \\ y \left (t \right ) &= -\frac {18^{\frac {1}{3}} \left (-t \right )^{\frac {4}{3}} \left (i \sqrt {3}-1\right )}{16} \\ y \left (t \right ) &= c_{1} \left (3 c_{1}^{3}+t \right ) \\ \end{align*}

Problem 14428

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

Problem 14429

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

Problem 14430

ODE	\[ \boxed {y-t y^{\prime }+4 {y^{\prime }}^{2}=0} \]
program solution	\[ y = -4 c_{1}^{2}+c_{1} t \] Verified OK. \[ y = \frac {t^{2}}{16} \] Verified OK.
Maple solution	\begin{align*} y \left (t \right ) &= \frac {t^{2}}{16} \\ y \left (t \right ) &= c_{1} \left (-4 c_{1} +t \right ) \\ \end{align*}

Problem 14431

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

Problem 14432

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

Problem 14433

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

Problem 14434

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

Problem 14435

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

Problem 14436

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

Problem 14437

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

Problem 14438

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

Problem 14439

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

Problem 14440

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

Problem 14441

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

Problem 14442

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

Problem 14443

ODE	\[ \boxed {y^{\prime }+\frac {y}{-2+t}=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{-c_{1}}}{-2+t} \] Warning, solution could not be verified
Maple solution	\[ y \left (t \right ) = 0 \]

Problem 14444

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

Problem 14445

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

Problem 14446

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

Problem 14447

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

Problem 14448

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

Problem 14449

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

Problem 14450

ODE	\[ \boxed {3 t^{2} y^{\prime \prime }-5 t y^{\prime }-3 y=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = 1, y^{\prime }\left (1\right ) = {\frac {17}{3}}\right ] \end {align*}
program solution	\[ y = \frac {9 t^{\frac {10}{3}}-4}{5 t^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {4}{5 t^{\frac {1}{3}}}+\frac {9 t^{3}}{5} \]

Problem 14451

ODE	\[ \boxed {t^{2} y^{\prime \prime }+7 t y^{\prime }-7 y=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 2, y^{\prime }\left (1\right ) = -22] \end {align*}
program solution	\[ y = -\frac {t^{8}-3}{t^{7}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {3}{t^{7}}-t \]

Problem 14452

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

Problem 14453

ODE	\[ \boxed {y^{\prime \prime }+10 y^{\prime }+24 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-6 t}+\frac {c_{2} {\mathrm e}^{-4 t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{-6 t}+c_{2} {\mathrm e}^{-4 t} \]

Problem 14454

ODE	\[ \boxed {y^{\prime \prime }+16 y=0} \]
program solution	\[ y = c_{1} \cos \left (4 t \right )+\frac {c_{2} \sin \left (4 t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \sin \left (4 t \right )+c_{2} \cos \left (4 t \right ) \]

Problem 14455

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+18 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-3 t} \cos \left (3 t \right )+\frac {c_{2} {\mathrm e}^{-3 t} \sin \left (3 t \right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-3 t} \left (c_{1} \sin \left (3 t \right )+\cos \left (3 t \right ) c_{2} \right ) \]

Problem 14456

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

Problem 14457

ODE	\[ \boxed {y^{\prime \prime }-5 y^{\prime }+6 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{3 t} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{3 t}-c_{2} {\mathrm e}^{3 t} {\mathrm e}^{-t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{2 t} \]

Problem 14458

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+8 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-2 t} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}-\frac {c_{2} {\mathrm e}^{-4 t}}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-2 t} c_{1} +c_{2} {\mathrm e}^{-4 t} \]

Problem 14459

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{2 t} \end {align*} With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = t \,{\mathrm e}^{2 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{2 t} t \]

Problem 14460

ODE	\[ \boxed {y^{\prime \prime }+10 y^{\prime }+25 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= {\mathrm e}^{-5 t} \end {align*}
program solution	\[ y = c_{1} {\mathrm e}^{-5 t}+c_{2} t \,{\mathrm e}^{-5 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-5 t} \left (c_{2} t +c_{1} \right ) \]

Problem 14461

ODE	\[ \boxed {y^{\prime \prime }+9 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \cos \left (3 t \right ) \end {align*} With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = -4] \end {align*}
program solution	\[ y = \cos \left (3 t \right )-\frac {4 \sin \left (3 t \right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \cos \left (3 t \right )-\frac {4 \sin \left (3 t \right )}{3} \]

Problem 14462

ODE	\[ \boxed {y^{\prime \prime }+49 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (7 t \right ) \end {align*}
program solution	\[ y = \sin \left (7 t \right ) c_{1} -\frac {c_{2} \sin \left (7 t \right ) \cot \left (7 t \right )}{7} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \sin \left (7 t \right )+c_{2} \cos \left (7 t \right ) \]

Problem 14463

ODE	\[ \boxed {t^{2} y^{\prime \prime }+4 t y^{\prime }-4 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{t^{4}} \end {align*}
program solution	\[ y = \frac {c_{1}}{t^{4}}+\frac {c_{2} t}{5} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} t^{5}+c_{2}}{t^{4}} \]

Problem 14464

ODE	\[ \boxed {t^{2} y^{\prime \prime }+6 t y^{\prime }+6 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{t^{2}} \end {align*}
program solution	\[ y = \frac {c_{1}}{t^{2}}-\frac {c_{2}}{t^{3}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{2} t +c_{1}}{t^{3}} \]

Problem 14465

ODE	\[ \boxed {t^{2} y^{\prime \prime }+t y^{\prime }+\left (t^{2}-\frac {1}{4}\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\cos \left (t \right )}{\sqrt {t}} \end {align*}
program solution	\[ y = \frac {\cos \left (t \right ) c_{1}}{\sqrt {t}}+\frac {c_{2} \cos \left (t \right ) \tan \left (t \right )}{\sqrt {t}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )}{\sqrt {t}} \]

Problem 14466

ODE	\[ \boxed {t^{2} y^{\prime \prime }+3 t y^{\prime }+y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{t} \end {align*}
program solution	\[ y = \frac {c_{1}}{t}+\frac {c_{2} \ln \left (t \right )}{t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{2} \ln \left (t \right )+c_{1}}{t} \]

Problem 14467

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

Problem 14468

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

Problem 14469

ODE	\[ \boxed {4 t^{2} y^{\prime \prime }+4 t y^{\prime }+\left (36 t^{2}-1\right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {1}{\sqrt {t}} \end {align*}
program solution	\[ y = \frac {c_{1}}{\sqrt {t}}+c_{2} \sqrt {t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (3 t \right )+\cos \left (3 t \right ) c_{2}}{\sqrt {t}} \]

Problem 14470

ODE	\[ \boxed {t y^{\prime \prime }+2 y^{\prime }+16 y t=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \left (4 t \right )}{t} \end {align*}
program solution	\[ y = \frac {\sin \left (4 t \right ) c_{1}}{t}-\frac {c_{2} \sin \left (4 t \right ) \cot \left (4 t \right )}{4 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {c_{1} \sin \left (4 t \right )+c_{2} \cos \left (4 t \right )}{t} \]

Problem 14471

ODE	\[ \boxed {y^{\prime \prime }+b \left (t \right ) y^{\prime }+c \left (t \right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \left (t \right )}{t^{2}} \end {align*}
program solution	\[ y = \frac {\sin \left (t \right ) c_{1}}{t^{2}}+\frac {c_{2} \sin \left (t \right ) \left (\int \csc \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{4}d t \right )}{t^{2}} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 14472

ODE	\[ \boxed {y^{\prime \prime }+b \left (t \right ) y^{\prime }+c \left (t \right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\cos \left (t \right )}{t^{3}} \end {align*} With initial conditions \begin {align*} [y \left (\pi \right ) = 0, y^{\prime }\left (2 \pi \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ \text {No solution found} \]

Problem 14473

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

Problem 14474

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }-12 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 t}+\frac {c_{2} {\mathrm e}^{6 t}}{8} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{6 t} c_{1} +c_{2} {\mathrm e}^{-2 t} \]

Problem 14475

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

Problem 14476

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

Problem 14477

ODE	\[ \boxed {y^{\prime \prime }+8 y^{\prime }+12 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-6 t}+\frac {c_{2} {\mathrm e}^{-2 t}}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{-6 t}+c_{2} {\mathrm e}^{-2 t} \]

Problem 14478

ODE	\[ \boxed {y^{\prime \prime }+5 y^{\prime }+y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}}+\frac {c_{2} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{21} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (-5+\sqrt {21}\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (5+\sqrt {21}\right ) t}{2}} \]

Problem 14479

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

Problem 14480

ODE	\[ \boxed {4 y^{\prime \prime }+9 y=0} \]
program solution	\[ y = c_{1} \cos \left (\frac {3 t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {3 t}{2}\right )}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \sin \left (\frac {3 t}{2}\right )+c_{2} \cos \left (\frac {3 t}{2}\right ) \]

Problem 14481

ODE	\[ \boxed {y^{\prime \prime }+16 y=0} \]
program solution	\[ y = c_{1} \cos \left (4 t \right )+\frac {c_{2} \sin \left (4 t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \sin \left (4 t \right )+c_{2} \cos \left (4 t \right ) \]

Problem 14482

ODE	\[ \boxed {y^{\prime \prime }+8 y=0} \]
program solution	\[ y = c_{1} \cos \left (2 \sqrt {2}\, t \right )+\frac {c_{2} \sqrt {2}\, \sin \left (2 \sqrt {2}\, t \right )}{4} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} \sin \left (2 \sqrt {2}\, t \right )+c_{2} \cos \left (2 \sqrt {2}\, t \right ) \]

Problem 14483

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

Problem 14484

ODE	\[ \boxed {4 y^{\prime \prime }+21 y^{\prime }+5 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-5 t}+\frac {4 c_{2} {\mathrm e}^{-\frac {t}{4}}}{19} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{1} {\mathrm e}^{-\frac {t}{4}}+c_{2} {\mathrm e}^{-5 t} \]

Problem 14485

ODE	\[ \boxed {7 y^{\prime \prime }+4 y^{\prime }-3 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-t}+\frac {7 c_{2} {\mathrm e}^{\frac {3 t}{7}}}{10} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (c_{1} {\mathrm e}^{\frac {10 t}{7}}+c_{2} \right ) {\mathrm e}^{-t} \]

Problem 14486

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

Problem 14487

ODE	\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{3 t}+c_{2} {\mathrm e}^{3 t} t \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{3 t} \left (c_{2} t +c_{1} \right ) \]

Problem 14488

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

Problem 14489

ODE	\[ \boxed {3 y^{\prime \prime }-y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 7] \end {align*}
program solution	\[ y = 21 \,{\mathrm e}^{\frac {t}{3}}-21 \] Verified OK.
Maple solution	\[ y \left (t \right ) = -21+21 \,{\mathrm e}^{\frac {t}{3}} \]

Problem 14490

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime }-12 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = 7] \end {align*}
program solution	\[ y = \frac {19 \,{\mathrm e}^{3 t}}{7}+\frac {2 \,{\mathrm e}^{-4 t}}{7} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (19 \,{\mathrm e}^{7 t}+2\right ) {\mathrm e}^{-4 t}}{7} \]

Problem 14491

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+12 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -2] \end {align*}
program solution	\[ y = -11 \,{\mathrm e}^{4 t}+14 \,{\mathrm e}^{3 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -11 \,{\mathrm e}^{4 t}+14 \,{\mathrm e}^{3 t} \]

Problem 14492

ODE	\[ \boxed {2 y^{\prime \prime }-7 y^{\prime }-4 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {2 \,{\mathrm e}^{4 t}}{9}-\frac {2 \,{\mathrm e}^{-\frac {t}{2}}}{9} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {2 \,{\mathrm e}^{4 t}}{9}-\frac {2 \,{\mathrm e}^{-\frac {t}{2}}}{9} \]

Problem 14493

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+10 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 5] \end {align*}
program solution	\[ y = {\mathrm e}^{5 t} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{5 t} \]

Problem 14494

ODE	\[ \boxed {y^{\prime \prime }+36 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = -6] \end {align*}
program solution	\[ y = 2 \cos \left (6 t \right )-\sin \left (6 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\sin \left (6 t \right )+2 \cos \left (6 t \right ) \]

Problem 14495

ODE	\[ \boxed {y^{\prime \prime }+100 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 10] \end {align*}
program solution	\[ y = \cos \left (10 t \right )+\sin \left (10 t \right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = \sin \left (10 t \right )+\cos \left (10 t \right ) \]

Problem 14496

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

Problem 14497

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

Problem 14498

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

Problem 14499

ODE	\[ \boxed {y^{\prime \prime }+4 y^{\prime }+20 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 2, y^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ y = {\mathrm e}^{-2 t} \left (2 \cos \left (4 t \right )+\sin \left (4 t \right )\right ) \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{-2 t} \left (\sin \left (4 t \right )+2 \cos \left (4 t \right )\right ) \]

Problem 14500

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