12.9.2 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 : Monday, January 27, 2025 at 05:34:29 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*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 18

dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)-y(x)=4/x^2,x],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 23

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

\[ y(x)\to \frac {4}{3 x^2}+\frac {c_1}{x}+c_2 x \]