Question

Given the equation, a(second partial-∂ x/∂ t) + b(∂ x /∂ t-first partial)+ c x = 0 show that A ⅇⅈ ω t is a solution for certain values of ω (Be sure to specify all relevant values of ω). (Begin by substituting x = A ⅇⅈ ω t into the above equation and then solve for values of ω such that the equation is satisfied for all values of A and t

Answer #1

at2y’’ + bty’ + cy=0 (t>0,a,b,c are all real numbers)
equation obtained from substituting:
A) x=log(t)
B) t^2= e^x

(a) Separate the following partial differential equation into
two ordinary differential equations: e 5t t 6 Uxx + 7t 2 Uxt − 6t 2
Ut = 0. (b) Given the boundary values Ux (0,t) = 0 and U(2π,t) = 0,
for all t, write an eigenvalue problem in terms of X(x) that the
equation in (a) must satisfy. That is, state (ONLY) the resulting
eigenvalue problem that you would need to solve next. You do not
need to actually solve...

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...

x(t+2) = x(t+1) . x(t) , t >=0
Solve a transformed version of the difference equation. Think of
a transformation that takes multiplication to addition, apply this
transformation to above ewn, and solve the transformed system.
We are given x(0) = 1 and x(1) = 1

Given the second-order differential equation
y''(x) − xy'(x) + x^2 y(x) = 0
with initial conditions
y(0) = 0, y'(0) = 1.
(a) Write this equation as a system of 2 first order
differential equations.
(b) Approximate its solution by using the forward Euler
method.

x''+8x'+25x=10u(t).....x(0)=0....x'(0)=0 please solve this
differential equation and show all steps including the
characteristic equation.

x’+2x=sin(t), x(0)=1
solve the first order differential equation.

If x1(t) and x2(t) are solutions to the differential
equation
x"+bx'+cx = 0
1. Is x= x1+x2+c for a constant c always a solution?
(I think No, except for the case of c=0)
2. Is tx1 a solution? (t is a constant)
I have to show all works of the whole process, please
help me!

Suppose x=c1e−t+c2e^5t. Verify that x=c1e^−t+c2e^5t is a
solution to x′′−4x′−5x=0 by substituting it into the differential
equation. (Enter the terms in the order given. Enter c1 as c1 and
c2 as c2.)

(a) Separate the following partial differential equation into
two ordinary differential equations: Utt + 4Utx − 2U = 0. (b) Given
the boundary values U(0,t) = 0 and Ux (L,t) = 0, L > 0, for all
t, write an eigenvalue problem in terms of X(x) that the equation
in (a) must satisfy.

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