78.16.9 problem 9

Internal problem ID [18309]
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 : 9
Date solved : Thursday, March 13, 2025 at 11:53:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 26

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

\[ y = \sin \left (\frac {x}{2}\right ) c_{2} +\cos \left (\frac {x}{2}\right ) c_{1} +x^{4}-48 x^{2}+384 \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 33

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

\[ y(x)\to x^4-48 x^2+c_1 \cos \left (\frac {x}{2}\right )+c_2 \sin \left (\frac {x}{2}\right )+384 \]

✓ Sympy. Time used: 0.091 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{1} \sin {\left (\frac {x}{2} \right )} + C_{2} \cos {\left (\frac {x}{2} \right )} + x^{4} - 48 x^{2} + 384 \]