68.1.6 problem Problem 1.3(d)

Internal problem ID [14064]
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 : Monday, March 31, 2025 at 12:02:26 PM
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*}

✓ Maple. Time used: 0.036 (sec). Leaf size: 34

ode:=diff(diff(y(x),x),x)+y(x) = f(x); 
ic:=y(0) = 0, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 0.679 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-f(x) + y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- \int f{\left (x \right )} \sin {\left (x \right )}\, dx + \int \limits ^{0} f{\left (x \right )} \sin {\left (x \right )}\, dx\right ) \cos {\left (x \right )} + \left (\int f{\left (x \right )} \cos {\left (x \right )}\, dx - \int \limits ^{0} f{\left (x \right )} \cos {\left (x \right )}\, dx\right ) \sin {\left (x \right )} \]