12.9.10 problem 10

Internal problem ID [1766]
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 : 10
Date solved : Monday, January 27, 2025 at 05:34:36 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+2 x \left (x -1\right ) y^{\prime }+\left (x^{2}-2 x +2\right ) y&=x^{3} {\mathrm e}^{2 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: 0.007 (sec). Leaf size: 27

dsolve([x^2*diff(y(x),x$2)+2*x*(x-1)*diff(y(x),x)+(x^2-2*x+2)*y(x)=x^3*exp(2*x),x*exp(-x)],singsol=all)
 

\[ y = \frac {x \left ({\mathrm e}^{2 x}+9 c_1 x \,{\mathrm e}^{-x}+9 c_2 \,{\mathrm e}^{-x}\right )}{9} \]

✓ Solution by Mathematica

Time used: 0.046 (sec). Leaf size: 30

DSolve[x^2*D[y[x],{x,2}]+2*x*(x-1)*D[y[x],x]+(x^2-2*x+2)*y[x]==x^3*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{9} e^{-x} x \left (e^{3 x}+9 (c_2 x+c_1)\right ) \]