67.15.63 problem 22.13 (a)

Internal problem ID [16781]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 22. Method of undetermined coefficients. Additional exercises page 412
Problem number : 22.13 (a)
Date solved : Thursday, October 02, 2025 at 01:38:47 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (5\right )}+18 y^{\prime \prime \prime }+81 y^{\prime }&=x^{2} {\mathrm e}^{3 x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 52

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+18*diff(diff(diff(y(x),x),x),x)+81*diff(y(x),x) = x^2*exp(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-3 c_4 x -3 c_2 +c_3 \right ) \cos \left (3 x \right )}{9}+\frac {\left (9 x^{2}-18 x +10\right ) {\mathrm e}^{3 x}}{8748}+\frac {\left (3 c_3 x +3 c_1 +c_4 \right ) \sin \left (3 x \right )}{9}+c_5 \]

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 62

ode=D[y[x],{x,5}]+18*D[y[x],{x,3}]+81*D[y[x],x]==x^2*Exp[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\left (\frac {e^{3 K[1]} (2-3 K[1])^2}{2916}+\cos (3 K[1]) (c_1+c_2 K[1])+(c_3+c_4 K[1]) \sin (3 K[1])\right )dK[1]+c_5 \end{align*}

✓ Sympy. Time used: 0.244 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*exp(3*x) + 81*Derivative(y(x), x) + 18*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \frac {x^{2} e^{3 x}}{972} - \frac {x e^{3 x}}{486} + \left (C_{2} + C_{3} x\right ) \sin {\left (3 x \right )} + \left (C_{4} + C_{5} x\right ) \cos {\left (3 x \right )} + \frac {5 e^{3 x}}{4374} \]