6.19.13 problem section 9.3, problem 13

Internal problem ID [2160]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 13
Date solved : Tuesday, September 30, 2025 at 05:24:29 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }-3 y^{\prime \prime }-7 y^{\prime }+6 y&=-3 \,{\mathrm e}^{-x} \left (-8 x^{2}+8 x +12\right ) \end{align*}

✓ Maple. Time used: 0.076 (sec). Leaf size: 36

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+3*diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)-7*diff(y(x),x)+6*y(x) = -3*exp(-x)*(-8*x^2+8*x+12); 
dsolve(ode,y(x), singsol=all);
 

\[ y = 3 \left (x -1\right )^{2} {\mathrm e}^{-x}+c_2 \,{\mathrm e}^{-3 x}+c_3 \,{\mathrm e}^{-2 x}+\left (c_4 x +c_1 \right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 45

ode=D[y[x],{x,4}]+3*D[y[x],{x,3}]-3*D[y[x],{x,2}]-7*D[y[x],x]+6*y[x]==-3*Exp[-x]*(12+8*x-8*x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-3 x} \left (3 e^{2 x} (x-1)^2+c_2 e^x+e^{4 x} (c_4 x+c_3)+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.278 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-24*x**2 + 24*x + 36)*exp(-x) + 6*y(x) - 7*Derivative(y(x), x) - 3*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} e^{- 3 x} + C_{4} e^{- 2 x} + \left (C_{1} + C_{2} x\right ) e^{x} + 3 \left (x^{2} - 2 x + 1\right ) e^{- x} \]