64.13.17 problem 17

Internal problem ID [13476]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.5. The Cauchy-Euler Equation. Exercises page 169
Problem number : 17
Date solved : Monday, March 31, 2025 at 07:59:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+4 y&=2 x \ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+4*y(x) = 2*x*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (2 \ln \left (x \right )\right ) c_2 +\cos \left (2 \ln \left (x \right )\right ) c_1 +\frac {2 x \ln \left (x \right )}{5}-\frac {4 x}{25} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 69

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

\[ y(x)\to \cos (2 \log (x)) \int _1^x-\log (K[1]) \sin (2 \log (K[1]))dK[1]+\sin (2 \log (x)) \int _1^x\cos (2 \log (K[2])) \log (K[2])dK[2]+c_1 \cos (2 \log (x))+c_2 \sin (2 \log (x)) \]

✓ Sympy. Time used: 0.310 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} \sin {\left (2 \log {\left (x \right )} \right )} + C_{2} \cos {\left (2 \log {\left (x \right )} \right )} + \frac {2 x \log {\left (x \right )}}{5} - \frac {4 x}{25} \]