Internal problem ID [12918]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 6. Laplace transform. Section 6.6. page 624
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+3 y=\left (1-\operatorname {Heaviside}\left (-2+t \right )\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{10}} \sin \left (-2+t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1, y^{\prime }\left (0\right ) = 2] \end {align*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 178
dsolve([diff(y(t),t$2)+diff(y(t),t)+3*y(t)=(1-Heaviside(t-2))*exp(-(t-2)/10)*sin(t-2),y(0) = 1, D(y)(0) = 2],y(t), singsol=all)
\[ y = \frac {8000 \operatorname {Heaviside}\left (-2+t \right ) \left (\left (\cos \left (t \right )-\frac {191 \sin \left (t \right )}{80}\right ) \cos \left (2\right )+\frac {191 \left (\cos \left (t \right )+\frac {80 \sin \left (t \right )}{191}\right ) \sin \left (2\right )}{80}\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{10}}}{42881}+\frac {100 \left (11 \left (80 \cos \left (2\right )+191 \sin \left (2\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )-318 \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \left (\cos \left (2\right )-\frac {782 \sin \left (2\right )}{795}\right )\right ) {\mathrm e}^{\frac {1}{5}-\frac {t}{2}}}{471691}+\left (-\frac {4000}{42881}+\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}-i\right ) \left (-2+t \right )}+\left (-\frac {4000}{42881}-\frac {9550 i}{42881}\right ) {\mathrm e}^{\left (-\frac {1}{10}+i\right ) \left (-2+t \right )}+\frac {200 \operatorname {Heaviside}\left (-2+t \right ) \left (\left (-159 \sin \left (\sqrt {11}\right ) \sqrt {11}-440 \cos \left (\sqrt {11}\right )\right ) \cos \left (\frac {\sqrt {11}\, t}{2}\right )+\left (159 \cos \left (\sqrt {11}\right ) \sqrt {11}-440 \sin \left (\sqrt {11}\right )\right ) \sin \left (\frac {\sqrt {11}\, t}{2}\right )\right ) {\mathrm e}^{1-\frac {t}{2}}}{471691}+\frac {5 \,{\mathrm e}^{-\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{11}+{\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right ) \]
✓ Solution by Mathematica
Time used: 6.103 (sec). Leaf size: 243
DSolve[{y''[t]+y'[t]+8*y[t]==(1-UnitStep[t-2])*Exp[-(t-2)/10]*Sin[t-2],{y[0]==1,y'[0]==2}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {e^{-t/2} \left (-248000 e^{\frac {2 t}{5}+\frac {1}{5}} \cos (2-t)+5 \left (\sqrt {31} \left (483881-8 \sqrt [5]{e} (3295 \cos (2)-1782 \sin (2))\right ) \sin \left (\frac {\sqrt {31} t}{2}\right )-428420 e^{\frac {2 t}{5}+\frac {1}{5}} \sin (2-t)\right )+31 \cos \left (\frac {\sqrt {31} t}{2}\right ) \left (483881+100 \sqrt [5]{e} (80 \cos (2)+691 \sin (2))\right )\right )}{15000311} & t\leq 2 \\ \frac {e^{-t/2} \left (-248000 e \cos \left (\frac {1}{2} \sqrt {31} (t-2)\right )+5 \sqrt {31} \left (26360 e \sin \left (\frac {1}{2} \sqrt {31} (t-2)\right )+\left (483881-8 \sqrt [5]{e} (3295 \cos (2)-1782 \sin (2))\right ) \sin \left (\frac {\sqrt {31} t}{2}\right )\right )+31 \cos \left (\frac {\sqrt {31} t}{2}\right ) \left (483881+100 \sqrt [5]{e} (80 \cos (2)+691 \sin (2))\right )\right )}{15000311} & \text {True} \\ \end {array} \\ \end {array} \]