67.15.65 problem 22.13 (c)

Internal problem ID [16783]
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 (c)
Date solved : Thursday, October 02, 2025 at 01:38:48 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} \sin \left (3 x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 73

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

\[ y = \frac {\left (\left (-75 x^{2}-310 x +417\right ) {\mathrm e}^{3 x}-303750 c_4 x -303750 c_2 +101250 c_3 \right ) \cos \left (3 x \right )}{911250}+\frac {\left (-1575 x^{2}+3240 x -1693\right ) \sin \left (3 x \right ) {\mathrm e}^{3 x}}{2733750}+\frac {\left (3 c_3 x +3 c_1 +c_4 \right ) \sin \left (3 x \right )}{9}+c_5 \]

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 100

ode=D[y[x],{x,5}]+18*D[y[x],{x,3}]+81*D[y[x],x]==x^2*Exp[3*x]*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[1])+c_2 K[1] \cos (3 K[1])+c_3 \sin (3 K[1])+c_4 K[1] \sin (3 K[1])-\frac {e^{3 K[1]} \left (4 \cos (3 K[1]) \left (225 K[1]^2-270 K[1]+94\right )+\left (675 K[1]^2-1560 K[1]+932\right ) \sin (3 K[1])\right )}{455625}\right )dK[1]+c_5 \end{align*}

✓ Sympy. Time used: 0.512 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2*exp(3*x)*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)
 

\[ y{\left (x \right )} = C_{1} + \left (C_{2} + x \left (C_{3} - \frac {7 x e^{3 x}}{12150} + \frac {4 e^{3 x}}{3375}\right ) - \frac {1693 e^{3 x}}{2733750}\right ) \sin {\left (3 x \right )} + \left (C_{4} + x \left (C_{5} - \frac {x e^{3 x}}{12150} - \frac {31 e^{3 x}}{91125}\right ) + \frac {139 e^{3 x}}{303750}\right ) \cos {\left (3 x \right )} \]