49.16.9 problem 2(d)

Internal problem ID [7707]
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(d)
Date solved : Sunday, March 30, 2025 at 12:19:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 44

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

\[ y = \frac {c_2 \left (4 \pi -1\right ) x^{-2 \sqrt {\pi }}+c_1 \left (4 \pi -1\right ) x^{2 \sqrt {\pi }}-x}{4 \pi -1} \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 39

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

\[ y(x)\to c_2 x^{2 \sqrt {\pi }}+c_1 x^{-2 \sqrt {\pi }}+\frac {x}{1-4 \pi } \]

✓ Sympy. Time used: 0.332 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = \frac {C_{1}}{x^{2 \sqrt {\pi }}} + C_{2} x^{2 \sqrt {\pi }} + \frac {x}{1 - 4 \pi } \]