6.19.23 problem section 9.3, problem 23

Internal problem ID [2170]
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 23
Date solved : Tuesday, September 30, 2025 at 05:24:34 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }-3 y^{\prime \prime }+4 y^{\prime }+4 y&={\mathrm e}^{2 x} \left (18 x^{2}+33 x +13\right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 38

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-2*diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = exp(2*x)*(18*x^2+33*x+13); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x^{4}+x^{3}+6 c_4 x +x^{2}+6 c_2 \right ) {\mathrm e}^{2 x}}{6}+\left (c_3 x +c_1 \right ) {\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 58

ode=1*D[y[x],{x,4}]-2*D[y[x],{x,3}]-3*D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==Exp[2*x]*(13+33*x+18*x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{54} e^{2 x} \left (9 x^4+9 x^3+9 x^2+18 (-1+3 c_4) x+10+54 c_3\right )+e^{-x} (c_2 x+c_1) \end{align*}

✓ Sympy. Time used: 0.317 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-18*x**2 - 33*x - 13)*exp(2*x) + 4*y(x) + 4*Derivative(y(x), x) - 3*Derivative(y(x), (x, 2)) - 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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