76.13.47 problem 56

Internal problem ID [17558]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 56
Date solved : Monday, March 31, 2025 at 04:17:26 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+\frac {5 y}{4}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+5/4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 30

ode=x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+125/100*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_2 \cos \left (\frac {\log (x)}{2}\right )+c_1 \sin \left (\frac {\log (x)}{2}\right )}{x} \]

✓ Sympy. Time used: 0.178 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + 5*y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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