43.13.4 problem 1(d)

Internal problem ID [8966]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 121
Problem number : 1(d)
Date solved : Tuesday, September 30, 2025 at 06:00:43 PM
CAS classification : [_Laguerre]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 13

ode:=x*diff(diff(y(x),x),x)-(1+x)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{x}+c_1 x +c_1 \]

✓ Mathematica. Time used: 0.23 (sec). Leaf size: 78

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

\begin{align*} y(x)&\to \sqrt {x} \exp \left (\frac {1}{2} \left (2 \int _1^x\frac {K[1]-1}{2 K[1]}dK[1]+x+1\right )\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {K[1]-1}{2 K[1]}dK[1]\right )dK[2]+c_1\right ) \end{align*}

✗ Sympy

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

False