6.19.20 problem section 9.3, problem 20

Internal problem ID [2167]
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 20
Date solved : Tuesday, September 30, 2025 at 05:24:32 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime \prime }-2 y^{\prime \prime }-6 y^{\prime }-4 y&=-{\mathrm e}^{2 x} \left (15 x^{2}+28 x +4\right ) \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 39

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)-6*diff(y(x),x)-4*y(x) = -exp(2*x)*(15*x^2+28*x+4); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\left (x^{3}-6 c_2 -x \right ) {\mathrm e}^{3 x}-6 c_3 \cos \left (x \right )-6 c_4 \sin \left (x \right )-6 c_1 \right ) {\mathrm e}^{-x}}{6} \]

✓ Mathematica. Time used: 0.1 (sec). Leaf size: 65

ode=1*D[y[x],{x,4}]+1*D[y[x],{x,3}]-2*D[y[x],{x,2}]-6*D[y[x],x]-4*y[x]==-Exp[2*x]*(4+28*x+15*x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{90} e^{-x} \left (-15 e^{3 x} x^3+15 e^{3 x} x-11 e^{3 x}+90 c_4 e^{3 x}+90 c_2 \cos (x)+90 c_1 \sin (x)+90 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.300 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((15*x**2 + 28*x + 4)*exp(2*x) - 4*y(x) - 6*Derivative(y(x), x) - 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} - \frac {x^{3}}{6} + \frac {x}{6}\right ) e^{2 x} + \left (C_{2} + C_{3} \sin {\left (x \right )} + C_{4} \cos {\left (x \right )}\right ) e^{- x} \]