23.3.24 problem 24

Internal problem ID [5738]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 24
Date solved : Tuesday, September 30, 2025 at 02:02:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y+y^{\prime \prime }&=x \sin \left (x \right )^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

ode:=4*y(x)+diff(diff(y(x),x),x) = x*sin(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-8 x^{2}+128 c_2 +1\right ) \sin \left (2 x \right )}{128}+\frac {\left (-x +32 c_1 \right ) \cos \left (2 x \right )}{32}+\frac {x}{8} \]

✓ Mathematica. Time used: 0.137 (sec). Leaf size: 41

ode=4*y[x] + D[y[x],{x,2}] == x*Sin[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{128} \left (\left (-8 x^2+1+128 c_2\right ) \sin (2 x)+16 x-4 (x-32 c_1) \cos (2 x)\right ) \end{align*}

✓ Sympy. Time used: 0.794 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*sin(x)**2 + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x}{8} + \left (C_{1} - \frac {x}{32}\right ) \cos {\left (2 x \right )} + \left (C_{2} - \frac {x^{2}}{16}\right ) \sin {\left (2 x \right )} \]