12.9.18 problem 18

Internal problem ID [1774]
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 : 18
Date solved : Monday, January 27, 2025 at 05:34:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 15

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

\[ y = \frac {{\mathrm e}^{x} \left (c_1 x +c_2 \right )}{x} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 19

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

\[ y(x)\to \frac {e^x (c_2 x+c_1)}{x} \]