43.12.1 problem 1(c.1)

Internal problem ID [8960]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 108
Problem number : 1(c.1)
Date solved : Tuesday, September 30, 2025 at 06:00:34 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 13

ode:=diff(diff(y(x),x),x)+1/x*diff(y(x),x)-1/x^2*y(x) = 0; 
ic:=[y(1) = 1, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x}{2}+\frac {1}{2 x} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 17

ode=D[y[x],{x,2}]+1/x*D[y[x],x]-1/x^2*y[x]==0; 
ic={y[1]==1,Derivative[1][y][1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^2+1}{2 x} \end{align*}

✓ Sympy. Time used: 0.094 (sec). Leaf size: 10

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

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