68.15.24 problem 24

Internal problem ID [17750]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 24
Date solved : Thursday, October 02, 2025 at 02:27:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+4 y&=\frac {1}{x^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 24

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+4*y(x) = 1/x^2; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 76

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==1/x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos (2 \log (x)) \int _1^x-\frac {\sin (2 \log (K[1]))}{2 K[1]^3}dK[1]+\sin (2 \log (x)) \int _1^x\frac {\cos (2 \log (K[2]))}{2 K[2]^3}dK[2]+c_1 \cos (2 \log (x))+c_2 \sin (2 \log (x)) \end{align*}

✓ Sympy. Time used: 0.250 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + 4*y(x) - 1/x**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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