12.19.15 problem section 9.3, problem 15

Internal problem ID [2162]
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 15
Date solved : Saturday, March 29, 2025 at 11:49:45 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+8 y^{\prime \prime \prime }+24 y^{\prime \prime }+32 y^{\prime }&=-16 \,{\mathrm e}^{-2 x} \left (-x^{3}+x^{2}+x +1\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 56

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+8*diff(diff(diff(y(x),x),x),x)+24*diff(diff(y(x),x),x)+32*diff(y(x),x) = -16*exp(-2*x)*(-x^3+x^2+x+1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (-c_2 -c_3 \right ) \cos \left (2 x \right )+\left (c_2 -c_3 \right ) \sin \left (2 x \right )-4 x^{3}+4 x^{2}+4 x +4\right ) {\mathrm e}^{-2 x}}{4}-\frac {c_1 \,{\mathrm e}^{-4 x}}{4}+c_4 \]

✓ Mathematica. Time used: 0.342 (sec). Leaf size: 64

ode=D[y[x],{x,4}]+8*D[y[x],{x,3}]+24*D[y[x],{x,2}]+32*D[y[x],x]==-16*Exp[-2*x]*(1+x+x^2-x^3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} e^{-2 x} \left (-4 x^3+4 x^2+4 x-c_3 e^{-2 x}-(c_1+c_2) \cos (2 x)+(c_2-c_1) \sin (2 x)+4\right )+c_4 \]

✓ Sympy. Time used: 0.562 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-16*x**3 + 16*x**2 + 16*x + 16)*exp(-2*x) + 32*Derivative(y(x), x) + 24*Derivative(y(x), (x, 2)) + 8*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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