68.1.6 problem Problem 1.3(d)

Internal problem ID [14135]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.3(d)
Date solved : Tuesday, January 28, 2025 at 06:15:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=f \left (x \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: 0.265 (sec). Leaf size: 34

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

\[ y = \left (\int _{0}^{x}\cos \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \sin \left (x \right )-\left (\int _{0}^{x}\sin \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \right ) \cos \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.053 (sec). Leaf size: 77

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

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