Internal problem ID [12917]
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: 3.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+8 y=\left (1-\operatorname {Heaviside}\left (t -4\right )\right ) \cos \left (t -4\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 128
dsolve([diff(y(t),t$2)+diff(y(t),t)+8*y(t)=(1-Heaviside(t-4))*cos(t-4),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y = -\frac {9 \operatorname {Heaviside}\left (t -4\right ) \left (\left (\sin \left (2 \sqrt {31}\right ) \sqrt {31}-\frac {217 \cos \left (2 \sqrt {31}\right )}{9}\right ) \cos \left (\frac {\sqrt {31}\, t}{2}\right )-\frac {217 \sin \left (\frac {\sqrt {31}\, t}{2}\right ) \left (\frac {9 \cos \left (2 \sqrt {31}\right ) \sqrt {31}}{217}+\sin \left (2 \sqrt {31}\right )\right )}{9}\right ) {\mathrm e}^{-\frac {t}{2}+2}}{1550}-\frac {7 \left (\cos \left (4\right )-\frac {\sin \left (4\right )}{7}\right ) {\mathrm e}^{-\frac {t}{2}} \cos \left (\frac {\sqrt {31}\, t}{2}\right )}{50}-\frac {9 \left (\cos \left (4\right )+\frac {13 \sin \left (4\right )}{9}\right ) \sqrt {31}\, {\mathrm e}^{-\frac {t}{2}} \sin \left (\frac {\sqrt {31}\, t}{2}\right )}{1550}-\frac {7 \left (-1+\operatorname {Heaviside}\left (t -4\right )\right ) \left (\left (\cos \left (t \right )+\frac {\sin \left (t \right )}{7}\right ) \cos \left (4\right )-\frac {\sin \left (4\right ) \left (\cos \left (t \right )-7 \sin \left (t \right )\right )}{7}\right )}{50} \]
✓ Solution by Mathematica
Time used: 4.688 (sec). Leaf size: 207
DSolve[{y''[t]+y'[t]+8*y[t]==(1-UnitStep[t-4])*Cos[t-4],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {e^{-t/2} \left (\theta (4-t) \left (-31 e^{t/2} \sin (4-t)-9 \sqrt {31} e^2 \sin \left (\frac {1}{2} \sqrt {31} (t-4)\right )+217 e^{t/2} \cos (4-t)-217 e^2 \cos \left (\frac {1}{2} \sqrt {31} (t-4)\right )\right )+9 \sqrt {31} e^2 \sin \left (\frac {1}{2} \sqrt {31} (t-4)\right )-13 \sqrt {31} \sin (4) \sin \left (\frac {\sqrt {31} t}{2}\right )+217 e^2 \cos \left (\frac {1}{2} \sqrt {31} (t-4)\right )-217 \cos (4) \cos \left (\frac {\sqrt {31} t}{2}\right )-9 \sqrt {31} \cos (4) \sin \left (\frac {\sqrt {31} t}{2}\right )+31 \sin (4) \cos \left (\frac {\sqrt {31} t}{2}\right )\right )}{1550} \]