Internal problem ID [849]
Book: Elementary differential equations and boundary value problems, 11th ed., Boyce, DiPrima,
Meade
Section: Chapter 6.4, The Laplace Transform. Differential equations with discontinuous forcing
functions. page 268
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+y^{\prime }+\frac {5 y}{4}-t +\operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (t -\frac {\pi }{2}\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 63
dsolve([diff(y(t),t$2)+diff(y(t),t)+5/4*y(t)=t-Heaviside(t-Pi/2)*(t-Pi/2),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = -\frac {16}{25}-\frac {12 \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right ) \left (\cos \left (t \right )+\frac {4 \sin \left (t \right )}{3}\right ) {\mathrm e}^{-\frac {t}{2}+\frac {\pi }{4}}}{25}+\frac {2 \left (8-10 t +5 \pi \right ) \operatorname {Heaviside}\left (t -\frac {\pi }{2}\right )}{25}+\frac {4 \left (4 \cos \left (t \right )-3 \sin \left (t \right )\right ) {\mathrm e}^{-\frac {t}{2}}}{25}+\frac {4 t}{5} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 87
DSolve[{y''[t]+y'[t]+5/4*y[t]==t-UnitStep[t-Pi/2]*(t-Pi/2),{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to {cc} \{ & {cc} \frac {4}{25} \left (5 t+e^{-t/2} (4 \cos (t)-3 \sin (t))-4\right ) & 2 t\leq \pi \\ \frac {2}{25} e^{-t/2} \left (8 \cos (t)-6 \sin (t)-2 e^{\pi /4} (3 \cos (t)+4 \sin (t))+5 e^{t/2} \pi \right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}