12.9.11 problem 11

Internal problem ID [1767]
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 : 11
Date solved : Monday, January 27, 2025 at 05:34:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (2 x -1\right ) y^{\prime }+\left (x^{2}-x -1\right ) y&=x^{2} {\mathrm e}^{x} \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 24

dsolve([x^2*diff(y(x),x$2)-x*(2*x-1)*diff(y(x),x)+(x^2-x-1)*y(x)=x^2*exp(x),x*exp(x)],singsol=all)
 

\[ y = \frac {{\mathrm e}^{x} \left (3 c_1 \,x^{2}+x^{3}+3 c_2 \right )}{3 x} \]

✓ Solution by Mathematica

Time used: 0.034 (sec). Leaf size: 32

DSolve[x^2*D[y[x],{x,2}]-x*(2*x-1)*D[y[x],x]+(x^2-x-1)*y[x]==x^2*Exp[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {e^x \left (2 x^3+3 c_2 x^2+6 c_1\right )}{6 x} \]