Question

Find the general solution of u_{xx} − 3u_{xy} +
2u_{yy} = 0 using the the method of characteristics: let v
= y + 2x and w = y + x; define U(v, w) to be U(v, w) = U(y + 2x, y
+ x) = u(x, y); derive and solve a PDE for U(v, w); convert back to
u(x, y).

Answer #1

8. Find the solution of the following PDE:
utt − 9uxx = 0
u(0, t) = u(3π, t) = 0
u(x, 0) = sin(x/3)
ut (x, 0) = 4 sin(x/3) − 6 sin(x)
9. Find the solution of the following PDE:
utt − uxx = 0
u(0, t) = u(1, t) = 0
u(x, 0) = 0
ut(x, 0) = x(1 − x)
10. Find the solution of the following PDE:
(1/2t+1)ut − uxx = 0
u(0,t) = u(π,t) =...

Find the general Solution to the PDE X*Uxy +
Uy = 0 and find a particular solution that satisfies
U(x,0) = x5 + x - 68/x, and U(2,y) = 3y4

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

Solve the heat equation and find the steady state solution:
uxx = ut, 0 < x < 1, t > 0,
u(0,t) = T1, u(1,t) = T2, where T1
and T2 are distinct
constants, and u(x,0) = 0

Use the METHOD of REDUCTION OF ORDER to find the general
solution of the differential equation y"-4y=2 given that y1=e^-2x
is a solution for the associated differential equation. When
solving, use y=y1u and w=u'.

Solve (find the general solution) of the following
non-homogeneous DE using the annihilator method. y′′ + 8y′ +15y =
2sin(2x)

Use the Fourier transform to find the solution of the following
initial boundaryvalue Laplace equations
uxx + uyy = 0, −∞ < x < ∞ 0 < y < a,
u(x, 0) = f(x), u(x, a) = 0, −∞ < x < ∞
u(x, y) → 0 uniformlyiny as|x| → ∞.

Solve the wave equation Utt - C^2 Uxx = 0 with initial condtions
:
1) u(x,0) = log (1+x^2), Ut(x,0) = 4+x
2) U(x,0) = x^3 , Ut(x,0) =sinx
(PDE)

Solve the heat equation and find the steady state solution :
uxx=ut 0<x<1, t>0,
u(0,t)=T1, u(1,t)=T2, where T1 and T2 are
distinct constants, and u(x,0)=0

Consider the second order linear partial differential equation
a(x, y)uxx + 2b(x, y)uxy + c(x, y)uyy + d(x, y)ux + e(x, y)uy +
f(x, y)u = 0. (1)
a). Show that under a coordinate transformation ξ = ξ(x, y), η =
η(x, y) the PDE (1) transforms into the PDE α(ξ, η)uξξ + 2β(ξ,
η)uξη + γ(ξ, η)uηη + δ(ξ, η)uξ + (ξ, η)uη + ψ(ξ, η)u = 0, (2) where
α(ξ, η) = aξ2 x + 2bξxξy + cξ2...

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