12.19.31 problem section 9.3, problem 31
Internal
problem
ID
[2178]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
9
Introduction
to
Linear
Higher
Order
Equations.
Section
9.3.
Undetermined
Coefficients
for
Higher
Order
Equations.
Page
495
Problem
number
:
section
9.3,
problem
31
Date
solved
:
Saturday, March 29, 2025 at 11:50:04 PM
CAS
classification
:
[[_3rd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }+2 y^{\prime }-2 y&={\mathrm e}^{2 x} \left (\left (-x^{2}+5 x +27\right ) \cos \left (x \right )+\left (9 x^{2}+13 x +2\right ) \sin \left (x \right )\right ) \end{align*}
✓ Maple. Time used: 0.011 (sec). Leaf size: 180
ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)+2*diff(y(x),x)-2*y(x) = exp(2*x)*((-x^2+5*x+27)*cos(x)+(9*x^2+13*x+2)*sin(x));
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\sqrt {2}\, \int -\left (\sqrt {2}\, \cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right ) \left (\left (-9 x^{2}-13 x -2\right ) \sin \left (x \right )+\left (x^{2}-5 x -27\right ) \cos \left (x \right )\right ) {\mathrm e}^{2 x}d x \cos \left (\sqrt {2}\, x \right )}{6}-\frac {\sqrt {2}\, \int -\left (\sqrt {2}\, \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right ) \left (\left (-9 x^{2}-13 x -2\right ) \sin \left (x \right )+\left (x^{2}-5 x -27\right ) \cos \left (x \right )\right ) {\mathrm e}^{2 x}d x \sin \left (\sqrt {2}\, x \right )}{6}+c_2 \cos \left (\sqrt {2}\, x \right )+c_3 \sin \left (\sqrt {2}\, x \right )+\frac {\left (5 \left (-x^{2}+x +3\right ) \cos \left (x \right )+4 \sin \left (x \right ) \left (x^{2}+\frac {5}{2} x +\frac {7}{4}\right )\right ) {\mathrm e}^{2 x}}{3}+c_1 \,{\mathrm e}^{x}
\]
✓ Mathematica. Time used: 0.019 (sec). Leaf size: 60
ode=1*D[y[x],{x,3}]-1*D[y[x],{x,2}]+2*D[y[x],x]-2*y[x]==Exp[2*x]*((27+5*x-x^2)*Cos[1*x]+(2+13*x+9*x^2)*Sin[1*x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{2 x} \left (\left (-x^2+x+1\right ) \cos (x)+(2 x+1) \sin (x)\right )+c_3 e^x+c_1 \cos \left (\sqrt {2} x\right )+c_2 \sin \left (\sqrt {2} x\right )
\]
✓ Sympy. Time used: 0.855 (sec). Leaf size: 58
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-(-x**2 + 5*x + 27)*cos(x) - (9*x**2 + 13*x + 2)*sin(x))*exp(2*x) - 2*y(x) + 2*Derivative(y(x), x) - Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} e^{x} + C_{2} \sin {\left (\sqrt {2} x \right )} + C_{3} \cos {\left (\sqrt {2} x \right )} + \left (- x^{2} \cos {\left (x \right )} + 2 x \sin {\left (x \right )} + x \cos {\left (x \right )} + \sin {\left (x \right )} + \cos {\left (x \right )}\right ) e^{2 x}
\]