15.15.2 problem 2

Internal problem ID [3206]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 24, page 109
Problem number : 2
Date solved : Sunday, March 30, 2025 at 01:21:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (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: 0.067 (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]
 

\[ y(x)\to \frac {1}{12} \left (2 x^3-3 x+12 c_2\right ) \sin (x)+\left (\frac {x^2}{4}-\frac {1}{8}+c_1\right ) \cos (x) \]

✓ Sympy. Time used: 0.154 (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)
 

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