6.9.32 problem 32

Internal problem ID [1788]
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 : 32
Date solved : Tuesday, September 30, 2025 at 05:19:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (3 x -1\right ) y^{\prime \prime }-\left (3 x +2\right ) y^{\prime }-\left (6 x -8\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 18

ode:=(3*x-1)*diff(diff(y(x),x),x)-(3*x+2)*diff(y(x),x)-(6*x-8)*y(x) = 0; 
ic:=[y(0) = 2, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 2 \,{\mathrm e}^{2 x}-{\mathrm e}^{-x} x \]

✓ Mathematica. Time used: 0.088 (sec). Leaf size: 21

ode=(3*x-1)*D[y[x],{x,2}]-(3*x+2)*D[y[x],x]-(6*x-8)*y[x]==0; 
ic={y[0]==2,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 2 e^{2 x}-e^{-x} x \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((3*x - 1)*Derivative(y(x), (x, 2)) - (3*x + 2)*Derivative(y(x), x) - (6*x - 8)*y(x),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics)
 

False