88.22.2 problem 6

Internal problem ID [24162]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 160 (Laplace transform)
Problem number : 6
Date solved : Thursday, October 02, 2025 at 10:00:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method

✓ Maple. Time used: 0.047 (sec). Leaf size: 27

ode:=diff(diff(y(x),x),x)+y(x) = x*sin(x); 
dsolve(ode,y(x),method='laplace');
 

\[ y = -\frac {\cos \left (x \right ) \left (x^{2}-4 y \left (0\right )\right )}{4}+\frac {\sin \left (x \right ) \left (4 y^{\prime }\left (0\right )+x \right )}{4} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 34

ode=D[y[x],{x,2}]+y[x]==x*Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left (\left (-2 x^2+1+8 c_1\right ) \cos (x)+2 (x+4 c_2) \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.081 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \left (C_{1} - \frac {x^{2}}{4}\right ) \cos {\left (x \right )} + \left (C_{2} + \frac {x}{4}\right ) \sin {\left (x \right )} \]