89.21.8 problem 10

Internal problem ID [24780]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Exercises at page 154
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:47:57 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime \prime }-18 y^{\prime \prime }+81 y&=36 \,{\mathrm e}^{3 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 34

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-18*diff(diff(y(x),x),x)+81*y(x) = 36*exp(3*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 45

ode=D[y[x],{x,4}]-18*D[y[x],{x,2}]+81*y[x]== 36*Exp[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.101 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(81*y(x) - 36*exp(3*x) - 18*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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