89.23.23 problem 23

Internal problem ID [24827]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Miscellaneous Exercises at page 162
Problem number : 23
Date solved : Thursday, October 02, 2025 at 10:48:22 PM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 27

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

\[ y = \frac {\left (-2 c_1 +x \right ) \cos \left (x \right )}{2}+\frac {\left (-2 c_2 -3\right ) \sin \left (x \right )}{2}+c_3 x +c_4 \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to c_4 x+\frac {1}{2} (x-2 c_1) \cos (x)-(1+c_2) \sin (x)+c_3 \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 22

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

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