85.69.8 problem 7 (c)

Internal problem ID [22924]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. C Exercises at page 217
Problem number : 7 (c)
Date solved : Thursday, October 02, 2025 at 09:16:40 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 \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.088 (sec). Leaf size: 65

ode:=diff(diff(y(x),x),x)+y(x) = f(x); 
ic:=[y(0) = 0, y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sin \left (x \right ) \left (-\int _{0}^{1}\cos \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} +\int _{0}^{1}\sin \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \cot \left (1\right )\right )+\int _{0}^{x}\cos \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \sin \left (x \right )-\int _{0}^{x}\sin \left (\textit {\_z1} \right ) f \left (\textit {\_z1} \right )d \textit {\_z1} \cos \left (x \right ) \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 66

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

\begin{align*} y(x)&\to \sin (x) \int _1^x\cos (K[2]) f(K[2])dK[2]+\cos (x) \int _1^x-f(K[1]) \sin (K[1])dK[1]-\csc (1) \sin (1-x) \int _1^0-f(K[1]) \sin (K[1])dK[1] \end{align*}

✓ Sympy. Time used: 0.428 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = \left (C_{1} - \int f{\left (x \right )} \sin {\left (x \right )}\, dx\right ) \cos {\left (x \right )} + \left (C_{2} + \int f{\left (x \right )} \cos {\left (x \right )}\, dx\right ) \sin {\left (x \right )} \]