87.8.9 problem 9

Internal problem ID [23363]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 65
Problem number : 9
Date solved : Thursday, October 02, 2025 at 09:40:33 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x^{2} y^{\prime \prime }-{\mathrm e}^{x} y^{\prime }-2&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 50

ode:=x^2*diff(diff(y(x),x),x)-exp(x)*diff(y(x),x)-2 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 66

ode=x^2*D[y[x],{x,2}]-Exp[x]*D[y[x],x]-2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^xe^{\operatorname {ExpIntegralEi}(K[2])-\frac {e^{K[2]}}{K[2]}} \left (c_1+\int _1^{K[2]}\frac {2 e^{\frac {e^{K[1]}}{K[1]}-\operatorname {ExpIntegralEi}(K[1])}}{K[1]^2}dK[1]\right )dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 5.340 (sec). Leaf size: 70

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

\[ y{\left (x \right )} = C_{1} + C_{2} \int \frac {1}{\int \frac {e^{x} e^{\frac {e^{x}}{x}} e^{- \operatorname {Ei}{\left (x \right )}}}{x^{2}}\, dx}\, dx - 2 \int \frac {\int \frac {e^{\frac {e^{x}}{x}} e^{- \operatorname {Ei}{\left (x \right )}}}{x^{2}}\, dx}{\int \frac {e^{x} e^{\frac {e^{x}}{x}} e^{- \operatorname {Ei}{\left (x \right )}}}{x^{2}}\, dx}\, dx \]