78.25.1 problem 5

Internal problem ID [18461]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 9. Laplace transforms. Section 52. Convolutions and Abels Mechanical Problem. Problems at page 474
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 11:50:25 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+a^{2} y&=f \left (x \right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.394 (sec). Leaf size: 26

dsolve([diff(y(x),x$2)+a^2*y(x)=f(x),y(0) = 0, D(y)(0) = 0],y(x), singsol=all)
 

\[ y = -\frac {\int _{0}^{x}\sin \left (a \left (-x +\textit {\_U1} \right )\right ) f \left (\textit {\_U1} \right )d \textit {\_U1}}{a} \]

✓ Solution by Mathematica

Time used: 0.074 (sec). Leaf size: 103

DSolve[{D[y[x],{x,2}]+a^2*y[x]==f[x],{y[0]==0,Derivative[1][y][0] == 0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\sin (a x) \int _1^0\frac {\cos (a K[2]) f(K[2])}{a}dK[2]+\sin (a x) \int _1^x\frac {\cos (a K[2]) f(K[2])}{a}dK[2]+\cos (a x) \left (\int _1^x-\frac {f(K[1]) \sin (a K[1])}{a}dK[1]-\int _1^0-\frac {f(K[1]) \sin (a K[1])}{a}dK[1]\right ) \]