12.9.29 problem 29

Internal problem ID [1785]
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 : 29
Date solved : Monday, January 27, 2025 at 05:34:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{2}-2 x \right ) y^{\prime \prime }+\left (-x^{2}+2\right ) y^{\prime }+\left (2 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: 0.003 (sec). Leaf size: 14

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

\[ y = c_1 \,x^{2}+c_2 \,{\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.038 (sec). Leaf size: 18

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

\[ y(x)\to c_2 x^2+c_1 e^x \]