69.15.8 problem 439

Internal problem ID [18246]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.2 Homogeneous differential equations with constant coefficients. Exercises page 121
Problem number : 439
Date solved : Thursday, October 02, 2025 at 03:09:47 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\left (6\right )}+2 y^{\left (5\right )}+y^{\prime \prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)+2*diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+diff(diff(diff(diff(y(x),x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 23.333 (sec). Leaf size: 175

ode=D[y[x],{x,6}]+2*D[y[x],{x,5}]+D[y[x],{x,4}]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\int _1^{K[4]}\int _1^{K[3]}\int _1^{K[2]}e^{-K[1]} (c_1+c_2 K[1])dK[1]dK[2]dK[3]dK[4]+x (x (c_6 x+c_5)+c_4)+c_3\\ y(x)&\to \frac {c_1 \left (x^3-6 x^2+15 x-16\right )}{6 e}+c_1 e^{-x}+x (x (c_6 x+c_5)+c_4)+c_3\\ y(x)&\to \frac {c_2 \left (2 x^3-15 x^2+48 x-65\right )}{6 e}+c_2 e^{-x} (x+4)+x (x (c_6 x+c_5)+c_4)+c_3 \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 4)) + 2*Derivative(y(x), (x, 5)) + Derivative(y(x), (x, 6)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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