64.11.52 problem 52
Internal
problem
ID
[13423]
Book
:
Differential
Equations
by
Shepley
L.
Ross.
Third
edition.
John
Willey.
New
Delhi.
2004.
Section
:
Chapter
4,
Section
4.3.
The
method
of
undetermined
coefficients.
Exercises
page
151
Problem
number
:
52
Date
solved
:
Wednesday, March 05, 2025 at 09:57:56 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime \prime }+16 y&=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) \end{align*}
✓ Maple. Time used: 0.011 (sec). Leaf size: 108
ode:=diff(diff(diff(diff(y(x),x),x),x),x)+16*y(x) = x*exp(2^(1/2)*x)*sin(2^(1/2)*x)+exp(-2^(1/2)*x)*cos(2^(1/2)*x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\left (2 \sqrt {2}\, x +128 c_{3} +3\right ) \cos \left (\sqrt {2}\, x \right )+2 \sin \left (\sqrt {2}\, x \right ) \left (\sqrt {2}\, x +64 c_4 \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{128}-\frac {\left (\left (\sqrt {2}\, x^{2}-128 c_{1} -\frac {5 \sqrt {2}}{8}\right ) \cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right ) \left (\sqrt {2}\, x^{2}-3 x -128 c_{2} +\frac {5 \sqrt {2}}{8}\right )\right ) {\mathrm e}^{\sqrt {2}\, x}}{128}
\]
✓ Mathematica. Time used: 1.913 (sec). Leaf size: 435
ode=D[y[x],{x,4}]+16*y[x]==x*Exp[Sqrt[2]*x]*Sin[Sqrt[2]*x]+Exp[-Sqrt[2]*x]*Cos[Sqrt[2]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-\sqrt {2} x} \left (e^{2 \sqrt {2} x} \cos \left (\sqrt {2} x\right ) \int _1^x-\frac {e^{-2 \sqrt {2} K[1]} \left (\cos \left (\sqrt {2} K[1]\right )+\sin \left (\sqrt {2} K[1]\right )\right ) \left (\cos \left (\sqrt {2} K[1]\right )+e^{2 \sqrt {2} K[1]} K[1] \sin \left (\sqrt {2} K[1]\right )\right )}{16 \sqrt {2}}dK[1]+\cos \left (\sqrt {2} x\right ) \int _1^x\frac {\left (\cos \left (\sqrt {2} K[2]\right )-\sin \left (\sqrt {2} K[2]\right )\right ) \left (\cos \left (\sqrt {2} K[2]\right )+e^{2 \sqrt {2} K[2]} K[2] \sin \left (\sqrt {2} K[2]\right )\right )}{16 \sqrt {2}}dK[2]+\sin \left (\sqrt {2} x\right ) \int _1^x\frac {\left (\cos \left (\sqrt {2} K[3]\right )+\sin \left (\sqrt {2} K[3]\right )\right ) \left (\cos \left (\sqrt {2} K[3]\right )+e^{2 \sqrt {2} K[3]} K[3] \sin \left (\sqrt {2} K[3]\right )\right )}{16 \sqrt {2}}dK[3]+e^{2 \sqrt {2} x} \sin \left (\sqrt {2} x\right ) \int _1^x\frac {e^{-2 \sqrt {2} K[4]} \left (\cos \left (\sqrt {2} K[4]\right )-\sin \left (\sqrt {2} K[4]\right )\right ) \left (\cos \left (\sqrt {2} K[4]\right )+e^{2 \sqrt {2} K[4]} K[4] \sin \left (\sqrt {2} K[4]\right )\right )}{16 \sqrt {2}}dK[4]+c_1 e^{2 \sqrt {2} x} \cos \left (\sqrt {2} x\right )+c_2 \cos \left (\sqrt {2} x\right )+c_3 \sin \left (\sqrt {2} x\right )+c_4 e^{2 \sqrt {2} x} \sin \left (\sqrt {2} x\right )\right )
\]
✓ Sympy. Time used: 0.568 (sec). Leaf size: 102
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x*exp(sqrt(2)*x)*sin(sqrt(2)*x) + 16*y(x) + Derivative(y(x), (x, 4)) - exp(-sqrt(2)*x)*cos(sqrt(2)*x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (\left (C_{1} - \frac {\sqrt {2} x^{2}}{128}\right ) \cos {\left (\sqrt {2} x \right )} + \left (C_{2} - \frac {\sqrt {2} x^{2}}{128} + \frac {3 x}{128}\right ) \sin {\left (\sqrt {2} x \right )}\right ) e^{\sqrt {2} x} + \left (\left (C_{3} + \frac {\sqrt {2} x}{64}\right ) \sin {\left (\sqrt {2} x \right )} + \left (C_{4} + \frac {\sqrt {2} x}{64}\right ) \cos {\left (\sqrt {2} x \right )}\right ) e^{- \sqrt {2} x}
\]