85.67.17 problem 17

Internal problem ID [22908]
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 216
Problem number : 17
Date solved : Thursday, October 02, 2025 at 09:16:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }-4 y^{\prime }+y&=\ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 32

ode:=4*diff(diff(y(x),x),x)-4*diff(y(x),x)+y(x) = ln(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 50

ode=4*D[y[x],{x,2}]-4*D[y[x],x]+y[x]==Log[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.837 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - log(x) - 4*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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