12.9.15 problem 15

Internal problem ID [1771]
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 : 15
Date solved : Monday, January 27, 2025 at 05:34:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y&=-{\mathrm e}^{-x} \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: 0.007 (sec). Leaf size: 38

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

\[ y = -\operatorname {Ei}_{1}\left (2 x \right ) x^{2} {\mathrm e}^{x}+\frac {{\mathrm e}^{-x} \left (2 x -1\right )}{4}+{\mathrm e}^{x} \left (c_1 \,x^{2}+c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.077 (sec). Leaf size: 52

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

\[ y(x)\to e^x x^2 \operatorname {ExpIntegralEi}(-2 x)+\frac {1}{4} e^{-x} \left (2 c_2 e^{2 x} x^2+2 x+4 c_1 e^{2 x}-1\right ) \]