13.10.8 problem 8

Internal problem ID [2409]
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
Date solved : Monday, January 27, 2025 at 05:51:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y&=f \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.057 (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)
 

\[ y = \frac {\left (\int _{0}^{t}{\mathrm e}^{-\textit {\_z1}} f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) {\mathrm e}^{t}}{2}-\frac {\left (\int _{0}^{t}f \left (\textit {\_z1} \right ) {\mathrm e}^{\textit {\_z1}}d \textit {\_z1} \right ) {\mathrm e}^{-t}}{2} \]

✓ Solution by Mathematica

Time used: 0.050 (sec). Leaf size: 103

DSolve[{D[y[t],{t,2}]-y[t]==f[t],{y[0]==0,Derivative[1][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{-t} \left (-e^{2 t} \int _1^0\frac {1}{2} e^{-K[1]} f(K[1])dK[1]+e^{2 t} \int _1^t\frac {1}{2} e^{-K[1]} f(K[1])dK[1]+\int _1^t-\frac {1}{2} e^{K[2]} f(K[2])dK[2]-\int _1^0-\frac {1}{2} e^{K[2]} f(K[2])dK[2]\right ) \]