87.17.40 problem 41

Internal problem ID [23632]
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 127
Problem number : 41
Date solved : Thursday, October 02, 2025 at 09:43:39 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y&=x \ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 65

ode:=diff(diff(y(x),x),x)-3*y(x) = x*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\sqrt {3}\, x} c_2 +{\mathrm e}^{-\sqrt {3}\, x} c_1 -\frac {x \ln \left (x \right )}{3}-\frac {{\mathrm e}^{\sqrt {3}\, x} \sqrt {3}\, \operatorname {Ei}_{1}\left (\sqrt {3}\, x \right )}{18}+\frac {{\mathrm e}^{-\sqrt {3}\, x} \sqrt {3}\, \operatorname {Ei}_{1}\left (-\sqrt {3}\, x \right )}{18} \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 92

ode=D[y[x],{x,2}]-3*y[x]==x*Log[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{18} e^{-\sqrt {3} x} \left (\sqrt {3} e^{2 \sqrt {3} x} \operatorname {ExpIntegralEi}\left (-\sqrt {3} x\right )-\sqrt {3} \operatorname {ExpIntegralEi}\left (\sqrt {3} x\right )-6 e^{\sqrt {3} x} x \log (x)+18 c_1 e^{2 \sqrt {3} x}+18 c_2\right ) \end{align*}

✓ Sympy. Time used: 3.993 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*log(x) - 3*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {x \log {\left (x \right )}}{3} + \left (C_{1} - \frac {\sqrt {3} \operatorname {Ei}{\left (\sqrt {3} x \right )}}{18}\right ) e^{- \sqrt {3} x} + \left (C_{2} + \frac {\sqrt {3} \operatorname {Ei}{\left (- \sqrt {3} x \right )}}{18}\right ) e^{\sqrt {3} x} \]