64.11.48 problem 48
Internal
problem
ID
[13419]
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
:
48
Date
solved
:
Wednesday, March 05, 2025 at 09:56:41 PM
CAS
classification
:
[[_high_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime \prime }+3 y^{\prime \prime \prime }+4 y^{\prime \prime }+3 y^{\prime }+y&=x^{2} {\mathrm e}^{-x}+3 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \end{align*}
✓ Maple. Time used: 0.008 (sec). Leaf size: 73
ode:=diff(diff(diff(diff(y(x),x),x),x),x)+3*diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x)+3*diff(y(x),x)+y(x) = x^2*exp(-x)+3*exp(-1/2*x)*cos(1/2*3^(1/2)*x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {3 \left (x -\frac {2 c_{3}}{3}+\frac {1}{3}\right ) {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {\left (\left (x -5\right ) \sqrt {3}-2 c_4 \right ) {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {{\mathrm e}^{-x} \left (-24+x^{4}+4 x^{3}+12 \left (-2+c_{2} \right ) x +12 c_{1} \right )}{12}
\]
✓ Mathematica. Time used: 1.316 (sec). Leaf size: 336
ode=D[y[x],{x,4}]+3*D[y[x],{x,3}]+4*D[y[x],{x,2}]+3*D[y[x],x]+y[x]==x^2*Exp[-x]+3*Exp[-x/2]*Cos[Sqrt[3]/2*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-x} \left (\int _1^x-\left ((K[3]-1) \left (K[3]^2+3 e^{\frac {K[3]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[3]\right )\right )\right )dK[3]+x \int _1^x\left (K[4]^2+3 e^{\frac {K[4]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[4]\right )\right )dK[4]+e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x-\frac {e^{-\frac {K[1]}{2}} \left (K[1]^2+3 e^{\frac {K[1]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[1]\right )\right ) \left (\cos \left (\frac {1}{2} \sqrt {3} K[1]\right )+\sqrt {3} \sin \left (\frac {1}{2} \sqrt {3} K[1]\right )\right )}{\sqrt {3}}dK[1]+e^{x/2} \cos \left (\frac {\sqrt {3} x}{2}\right ) \int _1^x-\frac {e^{-\frac {K[2]}{2}} \left (K[2]^2+3 e^{\frac {K[2]}{2}} \cos \left (\frac {1}{2} \sqrt {3} K[2]\right )\right ) \left (\sqrt {3} \cos \left (\frac {1}{2} \sqrt {3} K[2]\right )-\sin \left (\frac {1}{2} \sqrt {3} K[2]\right )\right )}{\sqrt {3}}dK[2]+c_4 x+c_2 e^{x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 e^{x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+c_3\right )
\]
✓ Sympy. Time used: 0.888 (sec). Leaf size: 63
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2*exp(-x) + y(x) + 3*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)) + 3*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)) - 3*exp(-x/2)*cos(sqrt(3)*x/2),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x^{3}}{12} + \frac {x^{2}}{3}\right )\right ) e^{- x} + \left (\left (C_{3} - \frac {3 x}{2}\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{4} - \frac {\sqrt {3} x}{2}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}}
\]