43.10.3 problem 1(c)
Internal
problem
ID
[8947]
Book
:
An
introduction
to
Ordinary
Differential
Equations.
Earl
A.
Coddington.
Dover.
NY
1961
Section
:
Chapter
2.
Linear
equations
with
constant
coefficients.
Page
89
Problem
number
:
1(c)
Date
solved
:
Tuesday, September 30, 2025 at 06:00:25 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime \prime }+16 y&=\cos \left (x \right ) \end{align*}
✓ Maple. Time used: 0.011 (sec). Leaf size: 67
ode:=diff(diff(diff(diff(y(x),x),x),x),x)+16*y(x) = cos(x);
dsolve(ode,y(x), singsol=all);
\[
y = c_2 \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_1 \,{\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+c_4 \,{\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+c_3 \,{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )+\frac {\cos \left (x \right )}{17}
\]
✓ Mathematica. Time used: 0.205 (sec). Leaf size: 335
ode=D[y[x],{x,4}]+16*y[x]==Cos[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to e^{-\sqrt {2} x} \left (e^{2 \sqrt {2} x} \cos \left (\sqrt {2} x\right ) \int _1^x-\frac {e^{-\sqrt {2} K[1]} \cos (K[1]) \left (\cos \left (\sqrt {2} K[1]\right )+\sin \left (\sqrt {2} K[1]\right )\right )}{16 \sqrt {2}}dK[1]+\cos \left (\sqrt {2} x\right ) \int _1^x\frac {e^{\sqrt {2} K[2]} \cos (K[2]) \left (\cos \left (\sqrt {2} K[2]\right )-\sin \left (\sqrt {2} K[2]\right )\right )}{16 \sqrt {2}}dK[2]+\sin \left (\sqrt {2} x\right ) \int _1^x\frac {e^{\sqrt {2} K[3]} \cos (K[3]) \left (\cos \left (\sqrt {2} K[3]\right )+\sin \left (\sqrt {2} K[3]\right )\right )}{16 \sqrt {2}}dK[3]+e^{2 \sqrt {2} x} \sin \left (\sqrt {2} x\right ) \int _1^x\frac {e^{-\sqrt {2} K[4]} \cos (K[4]) \left (\cos \left (\sqrt {2} K[4]\right )-\sin \left (\sqrt {2} K[4]\right )\right )}{16 \sqrt {2}}dK[4]+c_1 e^{2 \sqrt {2} x} \cos \left (\sqrt {2} x\right )+c_2 \cos \left (\sqrt {2} x\right )+c_3 \sin \left (\sqrt {2} x\right )+c_4 e^{2 \sqrt {2} x} \sin \left (\sqrt {2} x\right )\right ) \end{align*}
✓ Sympy. Time used: 0.111 (sec). Leaf size: 65
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(16*y(x) - cos(x) + Derivative(y(x), (x, 4)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{1} \sin {\left (\sqrt {2} x \right )} + C_{2} \cos {\left (\sqrt {2} x \right )}\right ) e^{- \sqrt {2} x} + \left (C_{3} \sin {\left (\sqrt {2} x \right )} + C_{4} \cos {\left (\sqrt {2} x \right )}\right ) e^{\sqrt {2} x} + \frac {\cos {\left (x \right )}}{17}
\]