66.2.39 problem Problem 54

Internal problem ID [13859]
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 54
Date solved : Wednesday, March 05, 2025 at 10:19:51 PM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 46

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

\[ y = \left (-c_{3} x -2 c_4 -c_{1} \right ) \cos \left (x \right )+\left (-c_4 x -c_{2} +2 c_{3} \right ) \sin \left (x \right )+\frac {x^{3}}{6}+x c_5 +c_6 +\frac {{\mathrm e}^{x}}{4} \]

✓ Mathematica. Time used: 0.346 (sec). Leaf size: 58

ode=D[y[x],{x,6}]+2*D[y[x],{x,4}]+D[y[x],{x,2}]==x+Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {x^3}{6}+\frac {e^x}{4}+c_6 x-(c_2 x+c_1+2 c_4) \cos (x)+(-c_4 x+2 c_2-c_3) \sin (x)+c_5 \]

✓ Sympy. Time used: 0.139 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x - exp(x) + Derivative(y(x), (x, 2)) + 2*Derivative(y(x), (x, 4)) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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