6.9.2 problem 2

Internal problem ID [1758]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 05:19:06 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

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

\[ y = c_2 x +\frac {c_1}{x}+\frac {4}{3 x^{2}} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to \frac {4}{3 x^2}+\frac {c_1}{x}+c_2 x \end{align*}

✓ Sympy. Time used: 0.193 (sec). Leaf size: 15

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

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