35.7.24 problem 29

Internal problem ID [6206]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 7. Other second-Order equations. page 435
Problem number : 29
Date solved : Sunday, March 30, 2025 at 10:43:04 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 16

ode:=x^2*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 x \,{\mathrm e}^{\frac {1}{x}}+c_1 x +c_1 \]

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 26

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*Derivative(y(x), (x, 2)) + y(x))/(x + 1) cannot be solved by the factorable group method