44.14.22 problem 3(f)

Internal problem ID [9346]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number : 3(f)
Date solved : Tuesday, September 30, 2025 at 06:16:45 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

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

\[ y = -\frac {3}{8}-\frac {{\mathrm e}^{-2 x} \operatorname {Ei}_{1}\left (-2 x \right ) \left (1+x \right )}{4}+\left (c_1 x +c_2 \right ) {\mathrm e}^{-2 x}+\frac {\left (x -1\right ) \ln \left (x \right )}{4} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 61

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

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

✓ Sympy. Time used: 2.664 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = \frac {x \log {\left (x \right )}}{4} + \left (C_{1} + x \left (C_{2} + \frac {\operatorname {Ei}{\left (2 x \right )}}{4}\right ) + \frac {\operatorname {Ei}{\left (2 x \right )}}{4}\right ) e^{- 2 x} - \frac {\log {\left (x \right )}}{4} - \frac {3}{8} \]