85.59.6 problem 6

Internal problem ID [22847]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 203
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:15:43 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=\ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 37

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

\[ y = {\mathrm e}^{x} c_2 +{\mathrm e}^{-2 x} c_1 -\frac {\ln \left (x \right )}{2}-\frac {{\mathrm e}^{x} \operatorname {Ei}_{1}\left (x \right )}{3}-\frac {{\mathrm e}^{-2 x} \operatorname {Ei}_{1}\left (-2 x \right )}{6} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 49

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

\begin{align*} y(x)&\to \frac {1}{6} \left (2 e^x \operatorname {ExpIntegralEi}(-x)+e^{-2 x} \operatorname {ExpIntegralEi}(2 x)-3 \log (x)+6 c_1 e^{-2 x}+6 c_2 e^x\right ) \end{align*}

✓ Sympy. Time used: 1.161 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = \left (C_{1} + \frac {\operatorname {Ei}{\left (- x \right )}}{3}\right ) e^{x} + \left (C_{2} + \frac {\operatorname {Ei}{\left (2 x \right )}}{6}\right ) e^{- 2 x} - \frac {\log {\left (x \right )}}{2} \]