66.2.11 problem Problem 11

Internal problem ID [13831]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 11
Date solved : Wednesday, March 05, 2025 at 10:18:46 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 53

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

\[ y = -\frac {\left (\left (\left (-16 c_{1} +\frac {1}{4}\right ) \cos \left (2 x \right )+x^{2}-16 c_4 \sin \left (2 x \right )\right ) {\mathrm e}^{2 x}-16 c_{3} {\mathrm e}^{4 x}-16 c_{2} -\frac {16 \,{\mathrm e}^{3 x}}{15}\right ) {\mathrm e}^{-2 x}}{16} \]

✓ Mathematica. Time used: 0.214 (sec). Leaf size: 50

ode=D[y[x],{x,4}]-16*y[x]==x^2-Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {x^2}{16}+\frac {e^x}{15}+c_1 e^{2 x}+c_3 e^{-2 x}+c_2 \cos (2 x)+c_4 \sin (2 x) \]

✓ Sympy. Time used: 0.121 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - 16*y(x) + exp(x) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{- 2 x} + C_{2} e^{2 x} + C_{3} \sin {\left (2 x \right )} + C_{4} \cos {\left (2 x \right )} - \frac {x^{2}}{16} + \frac {e^{x}}{15} \]