Question

3. Consider a second order linear homogeneous equation Ay'' + By' + Cy = 0 Suppose that e^at, e^bt and e^ct are solutions (where a, b, c are constants). A. Show that e^at + e^bt + e^ct is also a solution. B. Show that two of the numbers among a, b, c are equal.

Answer #1

a) The homogeneous and particular solutions of
the differential equation ay'' + by' + cy = f(x) are, respectively,
C1exp(x)+C2exp(-x) and 3x^3. Give the complete solution y(x) of the
differential equation.
b) If the force f(x) in the equation given in a)
is instead f(x) = f1(x) + f2(x) + f3(x), where f1(x), f2(x), and
f3(x) are generic forces, what would be the particular
solution?
c) The homogeneous solution of a forced oscillator
is cos(t) + sin(t), what is the...

Consider the second-order homogeneous linear equation
y''−6y'+9y=0.
(a) Use the substitution y=e^(rt) to attempt to find two
linearly independent solutions to the given equation.
(b) Explain why your work in (a) only results in one linearly
independent solution, y1(t).
(c) Verify by direct substitution that y2=te^(3t) is a solution
to y''−6y'+9y=0. Explain why this function is linearly independent
from y1 found in (a).
(d) State the general solution to the given equation

Consider the initial value problem, ay''+by'+cy=0, y(0)=d,
y'(0)=f where a,b,c,d and f are constants which one of the
following could be a solution to the initial value problem? Give
breif exlpanation to why the correct answer can be a solution, and
why the others can not possibly satisfy the equation.
a. sin(t)+e^t
b. cost+e^tsint
c. cost+1
d. e^tcost

Consider the following second order linear homogeneous ODE
?′′(?) − ??′(?) − ???(?) = ?, ?(?) = ?, ?′(?) = ??
Solve the equation using the characteristic equation
Transform the equation into a system (by setting
?1(?) = ?, ?2(?) = ?′ ) and solve it
again
State the nature of the critical point ?0, plot he portrait and
say if ?0 is stable, stable and attractive or unstable
(justify your answers)
Solve the equation using Laplace transform
Compare the...

Write down a homogeneous second-order linear differential
equation with constant coefficients whose solutions are:
a. e^-xcos(x) , e^-xsin(x)
b. x , e^x

Q.3 (Applications of Linear Second Order ODE): Consider the
‘equation of motion’ given by ODE d2x + ω2x = F0 cos(γt)
dt2
where F0 and ω ̸= γ are constants. Without worrying about those
constants, answer the questions (a)–(b).
(a) Show that the general solution of the given ODE is [2 pts]
x(t) := xc + xp = c1 cos(ωt) + c2 sin(ωt) + (F0 / ω2 − γ2)
cos(γt).
(b) Find the values of c1 and c2 if the...

A second order homogeneous linear differential equation has
odd-even parity. Prove that if one of its solutions is an even
function, the other can be constructed as an odd function.

Show that if two solutions of a second order homogeneous
differential equation with continuous coefficients on I have a
common zero then all their zeros are in common

Second-Order Linear Non-homogeneous with Constant Coefficients:
Find the general solution to the following differential equation,
using the Method of Undetermined Coefficients.
y''− 2y' + y = 4x + xe^x

Consider the differential equation x^2y′′ − 3xy′ − 5y = 0. Note
that this is not a constant coefficient differential equation, but
it is linear. The theory of linear differential equations states
that the dimension of the space of all homogeneous solutions equals
the order of the differential equation, so that a fundamental
solution set for this equation should have two linearly fundamental
solutions.
• Assume that y = x^r is a solution. Find the resulting
characteristic equation for r....

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