87.18.20 problem 20

Internal problem ID [23661]
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 135
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:44:00 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (1\right )&={\mathrm e} \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = exp(x)/x; 
ic:=[y(1) = exp(1), D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} \left (\ln \left (x \right ) x -2 x +3\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 18

ode=D[y[x],{x,2}]-2*D[y[x],x]+y[x]==1/x*Exp[x]; 
ic={y[1]==Exp[1],Derivative[1][y][1] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x (-2 x+x \log (x)+3) \end{align*}

✓ Sympy. Time used: 0.221 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \left (x \left (\log {\left (x \right )} - 2\right ) + 3\right ) e^{x} \]