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 : Saturday, March 29, 2025 at 11:39:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (x +1\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*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 38

ode:=x*diff(diff(y(x),x),x)-(2*x+1)*diff(y(x),x)+(1+x)*y(x) = -exp(-x); 
dsolve(ode,y(x), singsol=all);
 

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

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (x + 1)*y(x) - (2*x + 1)*Derivative(y(x), x) + exp(-x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x*y(x)*exp(x) + x*exp(x)*Derivative(y(x), (x, 2)) + y(x)*exp(x) + 1)*exp(-x)/(2*x + 1) cannot be solved by the factorable group method