43.16.6 problem 2(a)

Internal problem ID [8990]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 149
Problem number : 2(a)
Date solved : Tuesday, September 30, 2025 at 06:00:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+4 y&=1 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+4*y(x) = 1; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (2 \ln \left (x \right )\right ) c_2 +\cos \left (2 \ln \left (x \right )\right ) c_1 +\frac {1}{4} \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 25

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cos (2 \log (x))+c_2 \sin (2 \log (x))+\frac {1}{4} \end{align*}

✓ Sympy. Time used: 0.160 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + 4*y(x) - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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