67.15.64 problem 22.13 (b)

Internal problem ID [16782]
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 (b)
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} \sin \left (3 x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 59

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*sin(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (18 x^{4}-7776 c_4 x -66 x^{2}-7776 c_2 +2592 c_3 +19\right ) \cos \left (3 x \right )}{23328}+\frac {\left (-12 x^{3}+\left (13+1944 c_3 \right ) x +1944 c_1 +648 c_4 \right ) \sin \left (3 x \right )}{5832}+c_5 \]

✓ Mathematica. Time used: 0.302 (sec). Leaf size: 198

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

\begin{align*} y(x)&\to \int _1^x\left (c_1 \cos (3 K[5])+c_2 K[5] \cos (3 K[5])+\int _1^{K[5]}\frac {1}{54} K[1]^2 (3 \cos (3 K[1]) K[1]-\sin (3 K[1])) \sin (3 K[1])dK[1] \cos (3 K[5])+K[5] \int _1^{K[5]}-\frac {1}{36} K[2]^2 \sin (6 K[2])dK[2] \cos (3 K[5])+c_3 \sin (3 K[5])+c_4 K[5] \sin (3 K[5])+\sin (3 K[5]) \int _1^{K[5]}\frac {1}{54} K[3]^2 \sin (3 K[3]) (\cos (3 K[3])+3 K[3] \sin (3 K[3]))dK[3]+K[5] \sin (3 K[5]) \int _1^{K[5]}-\frac {1}{18} K[4]^2 \sin ^2(3 K[4])dK[4]\right )dK[5]+c_5 \end{align*}

✓ Sympy. Time used: 0.341 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*sin(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)
 

\[ y{\left (x \right )} = C_{1} + \left (C_{2} + x \left (C_{3} - \frac {x^{2}}{486}\right )\right ) \sin {\left (3 x \right )} + \left (C_{4} + x \left (C_{5} + \frac {x^{3}}{1296} - \frac {11 x}{3888}\right )\right ) \cos {\left (3 x \right )} \]