6.19.28 problem section 9.3, problem 28

Internal problem ID [2175]
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 28
Date solved : Tuesday, September 30, 2025 at 05:24:36 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-7 y^{\prime \prime \prime }+18 y^{\prime \prime }-20 y^{\prime }+8 y&={\mathrm e}^{2 x} \left (-5 x^{2}-8 x +3\right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 42

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-7*diff(diff(diff(y(x),x),x),x)+18*diff(diff(y(x),x),x)-20*diff(y(x),x)+8*y(x) = exp(2*x)*(-5*x^2-8*x+3); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\left (x^{5}-x^{4}-12 c_4 \,x^{2}-2 x^{3}-12 c_3 x -12 c_2 \right ) {\mathrm e}^{x}-12 c_1 \right ) {\mathrm e}^{x}}{12} \]

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 59

ode=1*D[y[x],{x,4}]-7*D[y[x],{x,3}]+18*D[y[x],{x,2}]-20*D[y[x],x]+8*y[x]==Exp[2*x]*(3-8*x-5*x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{12} e^{2 x} \left (-x^5+x^4+2 x^3+6 (-1+2 c_4) x^2+12 (1+c_3) x+12 (-1+c_2)\right )+c_1 e^x \end{align*}

✓ Sympy. Time used: 0.351 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((5*x**2 + 8*x - 3)*exp(2*x) + 8*y(x) - 20*Derivative(y(x), x) + 18*Derivative(y(x), (x, 2)) - 7*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + \left (C_{2} + x \left (C_{3} + x \left (C_{4} - \frac {x^{3}}{12} + \frac {x^{2}}{12} + \frac {x}{6}\right )\right )\right ) e^{x}\right ) e^{x} \]