Question

Partial differential equations

Solve using the method of characteristics

ut +1/2 ux + 3/2 vx = 0 , u(x,0) =cos(2x)

vt + 3/2 ux + 1/2 vx = 0 , v(x,0) = sin(2x)

Answer #1

Solve the below boundary value equation
1. Ut=2uxx o<x<pi 0<t
2. u(0,t) = ux(pi,t) 0<t
3. u(x,0) = 1-2x 0<x<pi

PDE
Solve using the method of characteristics
Plot the intial conditions and then solve the parial
differential equation
utt = c² uxx, -∞ < x < ∞, t > 0
u(x,0) = { 0 if x < -1 , 1-x² if -1≤ x ≤1, 0 if x > 0
ut(x,0) = 0

(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...

Using separation of variables to solve the heat equation, ut =
kuxx on the interval 0 < x < 1 with boundary conditions ux
(0, t ) = 0 and ux (1, t ) = 0, yields the general solution,
∞
u(x,t) = A0 + ?Ane−kλnt cos?nπx? (with λn = n2π2)
n=1DeterminethecoefficientsAn(n=0,1,2,...)whenu(x,0)=f(x)= 0,
1/2≤x<1 .

Solve the following Differential equations
a) x sin y dx + (x^2 + 1) cos y dy = 0

(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.

On this problem, be sure to discuss the dynamic behavior and
characteristics of the graphs.
3 (a) Solve AND find the characteristics curve for the eqtn
: ut + (x-t)ux = 0.
(b) Find the solution to the initial value problem u(0, x) =
e^{-x^2},
(c) and discuss its dynamic behavior.

Suppose u(t,x) and v(t,x ) is C^2 functions defined on R^2 that
satisfy the first-order system of PDE Ut=Vx,
Vt=Ux,
A.) Show that both U and V are classical solutions to the wave
equations Utt= Uxx.
Which result from multivariable calculus do you need to justify
the conclusion.
B)Given a classical sol. u(t,x) to the wave equation, can you
construct a function v(t,x) such that u(t,x), v(t,x)
form of sol. to the first order system.

Solve the following differential equations
1. cos(t)y' - sin(t)y = t^2
2. y' - 2ty = t
Solve the ODE
3. ty' - y = t^3 e^(3t), for t > 0
Compare the number of solutions of the following three initial
value problems for the previous ODE
4. (i) y(1) = 1 (ii) y(0) = 1 (iii) y(0) = 0
Solve the IVP, and find the interval of validity of the
solution
5. y' + (cot x)y = 5e^(cos x),...

Solve the following partial differential equation using
separation of variables method
[Marks 10]
?2?/??2− ???/?? +??/?? = ?.

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