44.10.13 problem 3(b)

Internal problem ID [9268]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.2. THE METHOD OF UNDETERMINED COEFFICIENTS. Page 67
Problem number : 3(b)
Date solved : Tuesday, September 30, 2025 at 06:15:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=2 \sin \left (3 x \right )+4 \sin \left (x \right )-26 \,{\mathrm e}^{-2 x}+27 x^{3} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 47

ode:=diff(diff(y(x),x),x)+9*y(x) = 2*sin(3*x)+4*sin(x)-26*exp(-2*x)+27*x^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.705 (sec). Leaf size: 116

ode=D[y[x],{x,2}]+9*y[x]==2*Sin[3*x]+4*Sin[x]-26*Exp[-2*x]+27*x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cos (3 x) \int _1^x\frac {1}{3} \left (-27 K[1]^3+26 e^{-2 K[1]}-4 \sin (K[1])-2 \sin (3 K[1])\right ) \sin (3 K[1])dK[1]+\sin (3 x) \int _1^x\frac {1}{3} \cos (3 K[2]) \left (27 K[2]^3-26 e^{-2 K[2]}+4 \sin (K[2])+2 \sin (3 K[2])\right )dK[2]+c_1 \cos (3 x)+c_2 \sin (3 x) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-27*x**3 + 9*y(x) - 4*sin(x) - 2*sin(3*x) + Derivative(y(x), (x, 2)) + 26*exp(-2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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