89.18.27 problem 27

Internal problem ID [24727]
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 : 27
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 (x \right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 37

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

\[ y = \frac {\sin \left (x \right )}{5}+c_1 \,{\mathrm e}^{x} \cos \left (x \right )+c_2 \,{\mathrm e}^{x} \sin \left (x \right )+c_3 \,{\mathrm e}^{-x} \cos \left (x \right )+c_4 \,{\mathrm e}^{-x} \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 48

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

\begin{align*} y(x)&\to \left (c_1 e^{-x}+c_4 e^x\right ) \cos (x)+\frac {1}{5} \left (5 c_2 e^{-x}+5 c_3 e^x+1\right ) \sin (x) \end{align*}

✓ Sympy. Time used: 0.083 (sec). Leaf size: 34

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

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