12.9.20 problem 20

Internal problem ID [1776]
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 : 20
Date solved : Monday, January 27, 2025 at 05:34:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \ln \left (x \right )^{2} y^{\prime \prime }-2 x \ln \left (x \right ) y^{\prime }+\left (2+\ln \left (x \right )\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\ln \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 12

dsolve([x^2*(ln(x))^2*diff(y(x),x$2)-2*x*ln(x)*diff(y(x),x)+(2+ln(x))*y(x)=0,ln(x)],singsol=all)
 

\[ y = \ln \left (x \right ) \left (c_2 x +c_1 \right ) \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 15

DSolve[x^2*Log[x]^2*D[y[x],{x,2}]-2*x*Log[x]*D[y[x],x]+(2+Log[x])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to (c_2 x+c_1) \log (x) \]