12.10.30 problem 30

Internal problem ID [1834]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 30
Date solved : Saturday, March 29, 2025 at 11:40:58 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 28

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

\[ y = \frac {\left (3 x^{2}-2 x +6\right ) {\mathrm e}^{2 x}}{6}+\frac {{\mathrm e}^{-x} x}{3} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 34

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

\[ y(x)\to \frac {1}{6} e^{-x} \left (e^{3 x} \left (3 x^2-2 x+6\right )+2 x\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-9*x**2*exp(2*x) - 6*x*y(x) + 6*x*exp(2*x) + 3*x*Derivative(y(x), (x, 2)) + 8*y(x) - exp(2*x) - Derivative(y(x), (x, 2)))/(3*x + 2) cannot be solved by the factorable group method