66.2.33 problem Problem 47

Internal problem ID [13932]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 47
Date solved : Tuesday, January 28, 2025 at 06:09:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{3} y^{\prime \prime }-x y^{\prime }+y&=0 \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: 16

dsolve([x^3*diff(y(x),x$2)-x*diff(y(x),x)+y(x)=0,x],singsol=all)
 

\[ y = \left (c_{1} {\mathrm e}^{-\frac {1}{x}}+c_{2} \right ) x \]

✓ Solution by Mathematica

Time used: 0.057 (sec). Leaf size: 20

DSolve[x^3*D[y[x],{x,2}]-x*D[y[x],x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \left (c_2 e^{-1/x}+c_1\right ) \]