Internal problem ID [10496]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY.
2015.
Section: Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page
156
Problem number: 6(g).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{\prime \prime }+\frac {2 x^{\prime }}{5}+2 x-1+\operatorname {Heaviside}\left (t -5\right )=0} \] With initial conditions \begin {align*} [x \left (0\right ) = 0, x^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 58
dsolve([diff(x(t),t$2)+4/10*diff(x(t),t)+2*x(t)=1-Heaviside(t-5),x(0) = 0, D(x)(0) = 0],x(t), singsol=all)
\[ x \left (t \right ) = \frac {\operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{-\frac {t}{5}+1} \left (\frac {\sin \left (-7+\frac {7 t}{5}\right )}{7}+\cos \left (-7+\frac {7 t}{5}\right )\right )}{2}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14}-\frac {\operatorname {Heaviside}\left (t -5\right )}{2}+\frac {1}{2} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 90
DSolve[{x''[t]+4/10*x'[t]+2*x[t]==1-UnitStep[t-5],{x[0]==0,x'[0]==0}},x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to {cc} \{ & {cc} \frac {1}{14} e^{-t/5} \left ((-7+7 e \cos (7)-e \sin (7)) \cos \left (\frac {7 t}{5}\right )+(-1+e (\cos (7)+7 \sin (7))) \sin \left (\frac {7 t}{5}\right )\right ) & t>5 \\ \frac {1}{2}-\frac {1}{14} e^{-t/5} \left (7 \cos \left (\frac {7 t}{5}\right )+\sin \left (\frac {7 t}{5}\right )\right ) & \text {True} \\ \\ \\ \\ \\ \end{align*}