78.16.19 problem 19

Internal problem ID [18319]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 23. Operator Methods for Finding Particular Solutions. Problems at page 169
Problem number : 19
Date solved : Thursday, March 13, 2025 at 11:54:13 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-8 y&=16 x^{2} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 40

ode:=diff(diff(diff(y(x),x),x),x)-8*y(x) = 16*x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -2 x^{2}+c_{1} {\mathrm e}^{2 x}+c_{2} {\mathrm e}^{-x} \cos \left (\sqrt {3}\, x \right )+c_3 \,{\mathrm e}^{-x} \sin \left (\sqrt {3}\, x \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 50

ode=D[y[x],{x,3}]-8*y[x]==16*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x} \left (-2 e^x x^2+c_1 e^{3 x}+c_2 \cos \left (\sqrt {3} x\right )+c_3 \sin \left (\sqrt {3} x\right )\right ) \]

✓ Sympy. Time used: 0.122 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-16*x**2 - 8*y(x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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