66.2.38 problem Problem 53

Internal problem ID [13866]
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 53
Date solved : Monday, March 31, 2025 at 08:15:36 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y^{\left (6\right )}-y&={\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 69

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

\[ y = \frac {\left (63 \,{\mathrm e}^{-\frac {x}{2}} c_3 +63 \,{\mathrm e}^{\frac {x}{2}} c_5 \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{63}+\frac {\left (63 \,{\mathrm e}^{-\frac {x}{2}} c_4 +63 \,{\mathrm e}^{\frac {x}{2}} c_6 \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{63}+c_2 \,{\mathrm e}^{-x}+c_1 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{2 x}}{63} \]

✓ Mathematica. Time used: 0.419 (sec). Leaf size: 85

ode=D[y[x],{x,6}]-y[x]==Exp[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {e^{2 x}}{63}+c_1 e^x+c_4 e^{-x}+e^{-x/2} \left (c_2 e^x+c_3\right ) \cos \left (\frac {\sqrt {3} x}{2}\right )+e^{-x/2} \left (c_6 e^x+c_5\right ) \sin \left (\frac {\sqrt {3} x}{2}\right ) \]

✓ Sympy. Time used: 0.261 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) - exp(2*x) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{5} e^{- x} + C_{6} e^{x} + \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} + \left (C_{3} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{\frac {x}{2}} + \frac {e^{2 x}}{63} \]