2.17.115 Problems 11401 to 11500

Problem 11401

ODE	\[ \boxed {7 t^{2} x^{\prime }-3 x=-2 t} \]
program solution	\[ x = -\frac {\left (2 \,\operatorname {expIntegral}_{1}\left (-\frac {3}{7 t}\right )-c_{1} \right ) {\mathrm e}^{-\frac {3}{7 t}}}{7} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {\left (2 \,\operatorname {expIntegral}_{1}\left (-\frac {3}{7 t}\right )-7 c_{1} \right ) {\mathrm e}^{-\frac {3}{7 t}}}{7} \]

Problem 11402

ODE	\[ \boxed {x^{\prime } x+x t=1} \]
program solution
Maple solution	\[ x \left (t \right ) = -\frac {\left (2^{\frac {2}{3}} t^{2}-4 \operatorname {RootOf}\left (\operatorname {AiryBi}\left (\textit {\_Z} \right ) 2^{\frac {1}{3}} c_{1} t +2^{\frac {1}{3}} t \operatorname {AiryAi}\left (\textit {\_Z} \right )-2 \operatorname {AiryBi}\left (1, \textit {\_Z}\right ) c_{1} -2 \operatorname {AiryAi}\left (1, \textit {\_Z}\right )\right )\right ) 2^{\frac {1}{3}}}{4} \]

Problem 11403

ODE	\[ \boxed {{x^{\prime }}^{2}+x t=\sqrt {1+t}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 11404

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

Problem 11405

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

Problem 11406

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

Problem 11407

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

Problem 11408

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

Problem 11409

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

Problem 11410

ODE	\[ \boxed {x^{\prime }+\frac {5 x}{t}=1+t} \] With initial conditions \begin {align*} [x \left (1\right ) = 1] \end {align*}
program solution	\[ x = \frac {6 t^{7}+7 t^{6}+29}{42 t^{5}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {t^{2}}{7}+\frac {t}{6}+\frac {29}{42 t^{5}} \]

Problem 11411

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

Problem 11412

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

Problem 11413

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

Problem 11414

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

Problem 11415

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

Problem 11416

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

Problem 11417

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

Problem 11418

ODE	\[ \boxed {x^{\prime }+\frac {{\mathrm e}^{-t} x}{t}=t} \] With initial conditions \begin {align*} [x \left (1\right ) = 0] \end {align*}
program solution	\[ x = \left (\int _{1}^{t}{\mathrm e}^{-\operatorname {expIntegral}_{1}\left (\textit {\_a} \right )} \textit {\_a} d \textit {\_a} \right ) {\mathrm e}^{\operatorname {expIntegral}_{1}\left (t \right )} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \left (\int _{1}^{t}\textit {\_z1} \,{\mathrm e}^{-\operatorname {expIntegral}_{1}\left (\textit {\_z1} \right )}d \textit {\_z1} \right ) {\mathrm e}^{\operatorname {expIntegral}_{1}\left (t \right )} \]

Problem 11419

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

Problem 11420

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

Problem 11421

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

Problem 11422

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

Problem 11423

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

Problem 11424

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

Problem 11425

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

Problem 11426

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

Problem 11427

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

Problem 11428

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

Problem 11429

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

Problem 11430

ODE	\[ \boxed {\frac {x}{t}+\left (x^{2}+\ln \left (t \right )\right ) x^{\prime }=-t^{3}} \]
program solution	\[ \frac {t^{4}}{4}+x \ln \left (t \right )+\frac {x^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} x \left (t \right ) &= \frac {\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {2}{3}}-4 \ln \left (t \right )}{2 \left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {1}{3}}} \\ x \left (t \right ) &= \frac {i \left (-\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {2}{3}}-4 \ln \left (t \right )\right ) \sqrt {3}-\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {2}{3}}+4 \ln \left (t \right )}{4 \left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {1}{3}}} \\ x \left (t \right ) &= \frac {i \left (\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {2}{3}}+4 \ln \left (t \right )\right ) \sqrt {3}-\left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {2}{3}}+4 \ln \left (t \right )}{4 \left (-3 t^{4}-12 c_{1} +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_{1} \right )^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 11431

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

Problem 11432

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

Problem 11433

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

Problem 11434

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

Problem 11435

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

Problem 11436

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

Problem 11437

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

Problem 11438

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

Problem 11439

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

Problem 11440

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

Problem 11441

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

Problem 11442

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

Problem 11443

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+4 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {\left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{2}\right )+15 \cos \left (\frac {\sqrt {15}\, t}{2}\right )\right ) {\mathrm e}^{-\frac {t}{2}}}{15} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {{\mathrm e}^{-\frac {t}{2}} \left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{2}\right )+15 \cos \left (\frac {\sqrt {15}\, t}{2}\right )\right )}{15} \]

Problem 11444

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

Problem 11445

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

Problem 11446

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

Problem 11447

ODE	\[ \boxed {2 x^{\prime \prime }+3 x^{\prime }+3 x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {\left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{4}\right )+5 \cos \left (\frac {\sqrt {15}\, t}{4}\right )\right ) {\mathrm e}^{-\frac {3 t}{4}}}{5} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {{\mathrm e}^{-\frac {3 t}{4}} \left (\sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{4}\right )+5 \cos \left (\frac {\sqrt {15}\, t}{4}\right )\right )}{5} \]

Problem 11448

ODE	\[ \boxed {\frac {x^{\prime \prime }}{2}+\frac {5 x^{\prime }}{6}+\frac {2 x}{9}=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 1, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {4 \,{\mathrm e}^{-\frac {t}{3}}}{3}-\frac {{\mathrm e}^{-\frac {4 t}{3}}}{3} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {4 \,{\mathrm e}^{-\frac {t}{3}}}{3}-\frac {{\mathrm e}^{-\frac {4 t}{3}}}{3} \]

Problem 11449

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

Problem 11450

ODE	\[ \boxed {x^{\prime \prime }+\frac {x^{\prime }}{8}+x=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 2, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = \frac {2 \left (\sqrt {255}\, \sin \left (\frac {\sqrt {255}\, t}{16}\right )+255 \cos \left (\frac {\sqrt {255}\, t}{16}\right )\right ) {\mathrm e}^{-\frac {t}{16}}}{255} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {2 \,{\mathrm e}^{-\frac {t}{16}} \left (\sqrt {255}\, \sin \left (\frac {\sqrt {255}\, t}{16}\right )+255 \cos \left (\frac {\sqrt {255}\, t}{16}\right )\right )}{255} \]

Problem 11451

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x=3 t^{3}-1} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}+3 t^{3}-9 t^{2}+17 \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +3 t^{3}-9 t^{2}+17 \]

Problem 11452

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

Problem 11453

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

Problem 11454

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x=t^{2} {\mathrm e}^{3 t}} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}-\frac {14 t \,{\mathrm e}^{3 t}}{169}+\frac {t^{2} {\mathrm e}^{3 t}}{13}+\frac {72 \,{\mathrm e}^{3 t}}{2197} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\frac {\left (169 t^{2}-182 t +72\right ) {\mathrm e}^{3 t}}{2197} \]

Problem 11455

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x=5 \sin \left (7 t \right )} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}-\frac {35 \cos \left (7 t \right )}{2353}-\frac {240 \sin \left (7 t \right )}{2353} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {240 \sin \left (7 t \right )}{2353}-\frac {35 \cos \left (7 t \right )}{2353} \]

Problem 11456

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x={\mathrm e}^{2 t} \cos \left (t \right )+t^{2}} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}+\frac {6 \,{\mathrm e}^{2 t} \cos \left (t \right )}{61}+\frac {5 \,{\mathrm e}^{2 t} \sin \left (t \right )}{61}-2 t +t^{2} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +t^{2}-2 t +\frac {\left (5 \sin \left (t \right )+6 \cos \left (t \right )\right ) {\mathrm e}^{2 t}}{61} \]

Problem 11457

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

Problem 11458

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x=\left (t +2\right ) \sin \left (\pi t \right )} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}-\frac {\pi t \cos \left (\pi t \right )}{\pi ^{4}-\pi ^{2}+1}+\frac {\left (-\pi ^{2}+1\right ) t \sin \left (\pi t \right )}{\pi ^{4}-\pi ^{2}+1}+\frac {\left (-4 \pi ^{5}+6 \pi ^{3}-2 \pi \right ) \cos \left (\pi t \right )}{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}}+\frac {\left (-2 \pi ^{6}+7 \pi ^{4}-5 \pi ^{2}+1\right ) \sin \left (\pi t \right )}{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {c_{1} {\mathrm e}^{-\frac {t}{2}} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+c_{2} {\mathrm e}^{-\frac {t}{2}} \left (\pi ^{4}-\pi ^{2}+1\right )^{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right )+\left (\left (-t -2\right ) \pi ^{6}+\left (2 t +7\right ) \pi ^{4}+\left (-2 t -5\right ) \pi ^{2}+t +1\right ) \sin \left (\pi t \right )-\left (\left (t +4\right ) \pi ^{4}+\left (-t -6\right ) \pi ^{2}+t +2\right ) \pi \cos \left (\pi t \right )}{\left (\pi ^{4}-\pi ^{2}+1\right )^{2}} \]

Problem 11459

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

Problem 11460

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x=5 \sin \left (2 t \right )+t \,{\mathrm e}^{t}} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}+\frac {t \,{\mathrm e}^{t}}{3}-\frac {{\mathrm e}^{t}}{3}-\frac {10 \cos \left (2 t \right )}{13}-\frac {15 \sin \left (2 t \right )}{13} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\frac {10 \cos \left (2 t \right )}{13}-\frac {15 \sin \left (2 t \right )}{13}+\frac {{\mathrm e}^{t} \left (t -1\right )}{3} \]

Problem 11461

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x=t^{3}+1-4 \cos \left (t \right ) t} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}+t^{3}-3 t^{2}+7-4 \sin \left (t \right ) t -4 \cos \left (t \right )+8 \sin \left (t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\left (-4 t +8\right ) \sin \left (t \right )+t^{3}-3 t^{2}-4 \cos \left (t \right )+7 \]

Problem 11462

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+x=-6+2 \,{\mathrm e}^{2 t} \sin \left (t \right )} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {3}}{3}-6-\frac {10 \,{\mathrm e}^{2 t} \cos \left (t \right )}{61}+\frac {12 \,{\mathrm e}^{2 t} \sin \left (t \right )}{61} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -6+\frac {2 \left (6 \sin \left (t \right )-5 \cos \left (t \right )\right ) {\mathrm e}^{2 t}}{61} \]

Problem 11463

ODE	\[ \boxed {x^{\prime \prime }+7 x=t \,{\mathrm e}^{3 t}} \]
program solution	\[ x = c_{1} \cos \left (\sqrt {7}\, t \right )+\frac {c_{2} \sqrt {7}\, \sin \left (\sqrt {7}\, t \right )}{7}+\frac {t \,{\mathrm e}^{3 t}}{16}-\frac {3 \,{\mathrm e}^{3 t}}{128} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \sin \left (\sqrt {7}\, t \right ) c_{2} +\cos \left (\sqrt {7}\, t \right ) c_{1} +\frac {\left (8 t -3\right ) {\mathrm e}^{3 t}}{128} \]

Problem 11464

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

Problem 11465

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

Problem 11466

ODE	\[ \boxed {x^{\prime \prime }-3 x^{\prime }-4 x=2 t^{2}} \]
program solution	\[ x = c_{1} {\mathrm e}^{-t}+\frac {c_{2} {\mathrm e}^{4 t}}{5}-\frac {t^{2}}{2}+\frac {3 t}{4}-\frac {13}{16} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{4 t} c_{2} +{\mathrm e}^{-t} c_{1} -\frac {t^{2}}{2}+\frac {3 t}{4}-\frac {13}{16} \]

Problem 11467

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

Problem 11468

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

Problem 11469

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime }+2 x=\sin \left (2 t \right ) t} \]
program solution	\[ x = c_{1} {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right )+\frac {2 c_{2} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) {\mathrm e}^{-\frac {t}{2}} \sqrt {7}}{7}-\frac {\cos \left (2 t \right ) t}{4}-\frac {\sin \left (2 t \right ) t}{4}-\frac {\cos \left (2 t \right )}{8}+\frac {\sin \left (2 t \right )}{2} \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {7}\, t}{2}\right ) c_{2} +{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {7}\, t}{2}\right ) c_{1} +\frac {\left (-2 t -1\right ) \cos \left (2 t \right )}{8}-\frac {\sin \left (2 t \right ) \left (t -2\right )}{4} \]

Problem 11470

ODE	\[ \boxed {x^{\prime \prime }-b x^{\prime }+x=\sin \left (2 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = -\frac {\left (b^{2}+b \sqrt {b^{2}-4}+6\right ) {\mathrm e}^{-\frac {\left (-b +\sqrt {b^{2}-4}\right ) t}{2}}+\left (b \sqrt {b^{2}-4}-b^{2}-6\right ) {\mathrm e}^{\frac {\left (b +\sqrt {b^{2}-4}\right ) t}{2}}-2 \left (\cos \left (2 t \right ) b -\frac {3 \sin \left (2 t \right )}{2}\right ) \sqrt {b^{2}-4}}{\sqrt {b^{2}-4}\, \left (4 b^{2}+9\right )} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (-\sqrt {b^{2}-4}\, b^{2}-b^{3}-6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{-\frac {\left (-b +\sqrt {b^{2}-4}\right ) t}{2}}+\left (\sqrt {b^{2}-4}\, b^{2}-b^{3}+6 \sqrt {b^{2}-4}+4 b \right ) {\mathrm e}^{\frac {\left (b +\sqrt {b^{2}-4}\right ) t}{2}}+2 \left (b^{3}-4 b \right ) \cos \left (2 t \right )+3 \left (-b^{2}+4\right ) \sin \left (2 t \right )}{4 b^{4}-7 b^{2}-36} \]

Problem 11471

ODE	\[ \boxed {x^{\prime \prime }-3 x^{\prime }-40 x=2 \,{\mathrm e}^{-t}} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ x = \frac {11 \,{\mathrm e}^{8 t}}{117}-\frac {{\mathrm e}^{-5 t}}{26}-\frac {{\mathrm e}^{-t}}{18} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {\left (-22 \,{\mathrm e}^{13 t}+13 \,{\mathrm e}^{4 t}+9\right ) {\mathrm e}^{-5 t}}{234} \]

Problem 11472

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

Problem 11473

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

Problem 11474

ODE	\[ \boxed {x^{\prime \prime }+\frac {x^{\prime }}{100}+4 x=\cos \left (2 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = -\frac {20000 \,{\mathrm e}^{-\frac {t}{200}} \sin \left (\frac {\sqrt {159999}\, t}{200}\right ) \sqrt {159999}}{159999}+50 \sin \left (2 t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {20000 \,{\mathrm e}^{-\frac {t}{200}} \sqrt {159999}\, \sin \left (\frac {\sqrt {159999}\, t}{200}\right )}{159999}+50 \sin \left (2 t \right ) \]

Problem 11475

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

Problem 11476

ODE	\[ \boxed {x^{\prime \prime }+3025 x=\cos \left (45 t \right )} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
program solution	\[ x = -\frac {\cos \left (55 t \right )}{1000}+\frac {\cos \left (45 t \right )}{1000} \] Verified OK.
Maple solution	\[ x \left (t \right ) = -\frac {\cos \left (55 t \right )}{1000}+\frac {\cos \left (45 t \right )}{1000} \]

Problem 11477

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

Problem 11478

ODE	\[ \boxed {x^{\prime \prime }-\frac {4 x}{t^{2}}=0} \]
program solution	\[ x = c_{1} t^{\frac {1}{2}-\frac {\sqrt {17}}{2}}+\frac {c_{2} \sqrt {17}\, t^{\frac {1}{2}+\frac {\sqrt {17}}{2}}}{17} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \sqrt {t}\, \left (t^{\frac {\sqrt {17}}{2}} c_{1} +t^{-\frac {\sqrt {17}}{2}} c_{2} \right ) \]

Problem 11479

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

Problem 11480

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

Problem 11481

ODE	\[ \boxed {t^{2} x^{\prime \prime }-7 x^{\prime } t +16 x=0} \]
program solution	\[ x = t^{4} c_{1} +c_{2} t^{4} \ln \left (t \right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = t^{4} \left (c_{2} \ln \left (t \right )+c_{1} \right ) \]

Problem 11482

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

Problem 11483

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

Problem 11484

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

Problem 11485

ODE	\[ \boxed {x^{\prime \prime }+t^{2} x^{\prime }=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 1] \end {align*}
program solution	\[ x = -\left (\int _{}^{0}{\mathrm e}^{-\frac {\textit {\_a}^{3}}{3}}d \textit {\_a} \right )+\int {\mathrm e}^{-\frac {t^{3}}{3}}d t \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {{\mathrm e}^{-\frac {t^{3}}{3}} \sqrt {t}\, \left (4 \,3^{\frac {5}{6}} \left (t^{3}\right )^{\frac {1}{6}}+9 \,{\mathrm e}^{\frac {t^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{6}, \frac {2}{3}, \frac {t^{3}}{3}\right )\right ) 3^{\frac {1}{6}} \left (\left \{\begin {array}{cc} \frac {1}{1-i \sqrt {3}} & t <0 \\ \frac {1}{2} & 0\le t \end {array}\right .\right )}{6} \]

Problem 11486

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

Problem 11487

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

Problem 11488

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

Problem 11489

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

Problem 11490

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

Problem 11491

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

Problem 11492

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

Problem 11493

ODE	\[ \boxed {t^{2} x^{\prime \prime }-3 x^{\prime } t +3 x=4 t^{7}} \]
program solution	\[ x = c_{1} t +\frac {1}{2} c_{2} t^{3}+\frac {1}{6} t^{7} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {\left (t^{6}+3 c_{1} t^{2}+6 c_{2} \right ) t}{6} \]

Problem 11494

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

Problem 11495

ODE	\[ \boxed {x^{\prime \prime }+x^{\prime } t +x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= {\mathrm e}^{-\frac {t^{2}}{2}} \end {align*}
program solution	\[ x = {\mathrm e}^{-\frac {t^{2}}{2}} c_{1} -\frac {i c_{2} {\mathrm e}^{-\frac {t^{2}}{2}} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )}{2} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \left (\operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right ) c_{1} +c_{2} \right ) {\mathrm e}^{-\frac {t^{2}}{2}} \]

Problem 11496

ODE	\[ \boxed {x^{\prime \prime }-x^{\prime } t +x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= t \end {align*}
program solution	\[ x = c_{1} t +c_{2} t \left (-\frac {{\mathrm e}^{\frac {t^{2}}{2}}}{t}-\frac {i \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )}{2}\right ) \] Verified OK.
Maple solution	\[ x \left (t \right ) = c_{2} {\mathrm e}^{\frac {t^{2}}{2}}+\frac {\left (i c_{2} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {2}\, t}{2}\right )+2 c_{1} \right ) t}{2} \]

Problem 11497

ODE	\[ \boxed {x^{\prime \prime }-2 a x^{\prime }+a^{2} x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= {\mathrm e}^{t a} \end {align*}
program solution	\[ x = {\mathrm e}^{t a} c_{1} +c_{2} {\mathrm e}^{t a} t \] Verified OK.
Maple solution	\[ x \left (t \right ) = {\mathrm e}^{a t} \left (c_{2} t +c_{1} \right ) \]

Problem 11498

ODE	\[ \boxed {x^{\prime \prime }-\frac {\left (t +2\right ) x^{\prime }}{t}+\frac {\left (t +2\right ) x}{t^{2}}=0} \] Given that one solution of the ode is \begin {align*} x_1 &= t \end {align*}
program solution	\[ x = c_{1} t +c_{2} t \,{\mathrm e}^{t} \] Verified OK.
Maple solution	\[ x \left (t \right ) = t \left (c_{1} +c_{2} {\mathrm e}^{t}\right ) \]

Problem 11499

ODE	\[ \boxed {t^{2} x^{\prime \prime }+x^{\prime } t +\left (t^{2}-\frac {1}{4}\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= \frac {\cos \left (t \right )}{\sqrt {t}} \end {align*}
program solution	\[ x = \frac {\cos \left (t \right ) c_{1}}{\sqrt {t}}+\frac {c_{2} \cos \left (t \right ) \tan \left (t \right )}{\sqrt {t}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right )}{\sqrt {t}} \]

Problem 11500

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