6.9.35 problem 35

Internal problem ID [1791]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 05:19:22 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} \left (x^{2}-4\right ) y^{\prime \prime }+4 x y^{\prime }+2 y&=x +2 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{x -2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {1}{3}} \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 27

ode:=(x^2-4)*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+2*y(x) = x+2; 
ic:=[y(0) = -1/3, D(y)(0) = -1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {x^{3}+6 x^{2}+24 x +8}{6 x^{2}-24} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 30

ode=(x^2-4)*D[y[x],{x,2}]+4*x*D[y[x],x]+2*y[x]==x+2; 
ic={y[1]==5/4,Derivative[1][y][1]==3/2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {-2 x^3-12 x^2+54 x+5}{48-12 x^2} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), x) - x + (x**2 - 4)*Derivative(y(x), (x, 2)) + 2*y(x) - 2,0) 
ics = {y(0): -1/3, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x*(-x*Derivative(y(x), (x, 2)) + 1) - 2*y(x) + 4*Derivative(y(x), (x, 2)) + 2)/(4*x) cannot be solved by the factorable group method