66.2.11 problem Problem 11

Internal problem ID [13839]
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 : Monday, March 31, 2025 at 08:14:49 AM
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: 0.010 (sec). Leaf size: 44

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

\[ y = \frac {{\mathrm e}^{x}}{15}-\frac {x^{2}}{16}-\frac {\cos \left (2 x \right )}{64}+c_1 \cos \left (2 x \right )+c_2 \,{\mathrm e}^{-2 x}+c_3 \,{\mathrm e}^{2 x}+c_4 \sin \left (2 x \right ) \]

✓ 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)
 

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