67.17.48 problem 48
Internal
problem
ID
[16869]
Book
:
Ordinary
Differential
Equations.
An
introduction
to
the
fundamentals.
Kenneth
B.
Howell.
second
edition.
CRC
Press.
FL,
USA.
2020
Section
:
Chapter
25.
Review
exercises
for
part
III.
page
447
Problem
number
:
48
Date
solved
:
Thursday, October 02, 2025 at 01:39:57 PM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} y^{\left (6\right )}-64 y&={\mathrm e}^{-2 x} \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 60
ode:=diff(diff(diff(diff(diff(diff(y(x),x),x),x),x),x),x)-64*y(x) = exp(-2*x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {{\mathrm e}^{-2 x} \left (-192 \left ({\mathrm e}^{3 x} c_3 +c_5 \,{\mathrm e}^{x}\right ) \cos \left (\sqrt {3}\, x \right )-192 \left ({\mathrm e}^{3 x} c_4 +c_6 \,{\mathrm e}^{x}\right ) \sin \left (\sqrt {3}\, x \right )-192 c_2 \,{\mathrm e}^{4 x}+x -192 c_1 \right )}{192}
\]
✓ Mathematica. Time used: 0.558 (sec). Leaf size: 324
ode=D[y[x],{x,6}]-64*y[x]==Exp[-2*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^x \cos \left (\sqrt {3} x\right ) \int _1^x\frac {1}{192} e^{-3 K[1]} \left (\cos \left (\sqrt {3} K[1]\right )+\sqrt {3} \sin \left (\sqrt {3} K[1]\right )\right )dK[1]+e^{-x} \cos \left (\sqrt {3} x\right ) \int _1^x\frac {1}{192} e^{-K[2]} \left (\sqrt {3} \sin \left (\sqrt {3} K[2]\right )-\cos \left (\sqrt {3} K[2]\right )\right )dK[2]+e^{-x} \sin \left (\sqrt {3} x\right ) \int _1^x-\frac {1}{192} e^{-K[3]} \left (\sqrt {3} \cos \left (\sqrt {3} K[3]\right )+\sin \left (\sqrt {3} K[3]\right )\right )dK[3]+e^x \sin \left (\sqrt {3} x\right ) \int _1^x\frac {1}{192} e^{-3 K[4]} \left (\sin \left (\sqrt {3} K[4]\right )-\sqrt {3} \cos \left (\sqrt {3} K[4]\right )\right )dK[4]-\frac {1}{192} e^{-2 x} x-\frac {e^{-2 x}}{768}+c_1 e^{2 x}+c_4 e^{-2 x}+c_2 e^x \cos \left (\sqrt {3} x\right )+c_3 e^{-x} \cos \left (\sqrt {3} x\right )+c_5 e^{-x} \sin \left (\sqrt {3} x\right )+c_6 e^x \sin \left (\sqrt {3} x\right ) \end{align*}
✓ Sympy. Time used: 0.169 (sec). Leaf size: 66
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-64*y(x) + Derivative(y(x), (x, 6)) - exp(-2*x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{6} e^{2 x} + \left (C_{1} - \frac {x}{192}\right ) e^{- 2 x} + \left (C_{2} \sin {\left (\sqrt {3} x \right )} + C_{3} \cos {\left (\sqrt {3} x \right )}\right ) e^{- x} + \left (C_{4} \sin {\left (\sqrt {3} x \right )} + C_{5} \cos {\left (\sqrt {3} x \right )}\right ) e^{x}
\]