Internal problem ID [1761]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.4, The method of variation of parameters. Page 154
Problem number: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }-y=f \left (t \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 0, y^{\prime }\left (0\right ) = 0] \end {align*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 39
dsolve([diff(y(t),t$2)-y(t)=f(t),y(0) = 0, D(y)(0) = 0],y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (\int _{0}^{t}{\mathrm e}^{-\textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) {\mathrm e}^{t}}{2}-\frac {\left (\int _{0}^{t}{\mathrm e}^{\textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) {\mathrm e}^{-t}}{2} \]
✓ Solution by Mathematica
Time used: 0.064 (sec). Leaf size: 103
DSolve[{y''[t]-y[t]==f[t],{y[0]==0,y'[0]==0}},y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{-t} \left (-e^{2 t} \int _1^0\frac {1}{2} e^{-K[1]} f(K[1])dK[1]+e^{2 t} \int _1^t\frac {1}{2} e^{-K[1]} f(K[1])dK[1]+\int _1^t-\frac {1}{2} e^{K[2]} f(K[2])dK[2]-\int _1^0-\frac {1}{2} e^{K[2]} f(K[2])dK[2]\right ) \]