89.22.9 problem 9

Internal problem ID [24797]
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. Exercises at page 161
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:48:06 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+4 y^{\prime }&=4 x^{3}+2 x \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 30

ode:=diff(diff(diff(y(x),x),x),x)+4*diff(y(x),x) = 4*x^3+2*x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{4}}{4}+\frac {\sin \left (2 x \right ) c_1}{2}-\frac {\cos \left (2 x \right ) c_2}{2}-\frac {x^{2}}{2}+c_3 \]

✓ Mathematica. Time used: 0.053 (sec). Leaf size: 38

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

\begin{align*} y(x)&\to \frac {1}{4} \left (x^4-2 x^2-2 c_2 \cos (2 x)+2 c_1 \sin (2 x)+4 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 27

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

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