85.63.1 problem 5

Internal problem ID [22867]
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 210
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:15:56 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)+y(x) = x^2*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (2 x^{3}+12 c_2 -3 x \right ) \sin \left (x \right )}{12}+\frac {\cos \left (x \right ) \left (x^{2}+4 c_1 \right )}{4} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \frac {1}{12} \left (2 x^3-3 x+12 c_2\right ) \sin (x)+\left (\frac {x^2}{4}-\frac {1}{8}+c_1\right ) \cos (x) \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = \left (C_{1} + \frac {x^{2}}{4}\right ) \cos {\left (x \right )} + \left (C_{2} + \frac {x^{3}}{6} - \frac {x}{4}\right ) \sin {\left (x \right )} \]