64.9.6 problem 6

Internal problem ID [13404]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.1. Basic theory of linear differential equations. Exercises page 124
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 05:41:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}-x^{2}\right ) y^{\prime \prime }-\left (x^{3}+2 x^{2}-2 x \right ) y^{\prime }+\left (2 x^{2}+2 x -2\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

Time used: 0.254 (sec). Leaf size: 96

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

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]-2}{2 (K[1]-1)}dK[1]-\frac {1}{2} \int _1^x\left (-\frac {2}{K[2]}-1+\frac {1}{1-K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3]+c_1\right ) \]