89.18.28 problem 28

Internal problem ID [24728]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Oral Exercises at page 146
Problem number : 28
Date solved : Thursday, October 02, 2025 at 10:47:29 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+4 y&=\sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+4*y(x) = sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sin \left (2 x \right )}{20}+c_1 \,{\mathrm e}^{x} \cos \left (x \right )+c_2 \,{\mathrm e}^{x} \sin \left (x \right )+c_3 \,{\mathrm e}^{-x} \cos \left (x \right )+c_4 \,{\mathrm e}^{-x} \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.232 (sec). Leaf size: 49

ode=D[y[x],{x,4}]+4*y[x]==Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-x} \left (c_3 e^{2 x}+c_2\right ) \sin (x)+\cos (x) \left (\frac {\sin (x)}{10}+c_1 e^{-x}+c_4 e^x\right ) \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )}\right ) e^{- x} + \left (C_{3} \sin {\left (x \right )} + C_{4} \cos {\left (x \right )}\right ) e^{x} + \frac {\sin {\left (2 x \right )}}{20} \]