76.13.51 problem 60

Internal problem ID [17562]
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 : 60
Date solved : Monday, March 31, 2025 at 04:17:34 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 29

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

\[ y = \frac {c_1 \sin \left (\frac {\sqrt {15}\, \ln \left (x \right )}{2}\right )+c_2 \cos \left (\frac {\sqrt {15}\, \ln \left (x \right )}{2}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 42

ode=x^2*D[y[x],{x,2}]+2*x*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_2 \cos \left (\frac {1}{2} \sqrt {15} \log (x)\right )+c_1 \sin \left (\frac {1}{2} \sqrt {15} \log (x)\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.189 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = \frac {C_{1} \sin {\left (\frac {\sqrt {15} \log {\left (x \right )}}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {15} \log {\left (x \right )}}{2} \right )}}{\sqrt {x}} \]