Question

Solve the PDE using the change of variables v = x, w = y/x

x^{2}u_{xx} + 2xyu_{xy} +
y^{2}u_{yy} = 4x^{2}

Answer #1

Solve the PDE using the change of variables v = x, w = y/x
x2uxx + 2xyuxy +
y2uyy = 4x2

(PDE)
Solve the ff boundary value problems using Laplace Equationnon
the square , omega= { o<x<phi, 0< y <phi}:
u(x,0) =0, u(x,phi) = 0 ; u(0,y)= siny , u(phi,y) =0

Find the general solution of uxx − 3uxy +
2uyy = 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).

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

Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...

Find numbers x and y so that w-x⋅u-y⋅v is perpendicular to both
u and v, where w=[-28,-25,39], u=[1,-4,2], and v=[7,3,2].

Use a change of variables to evaluate Z Z R (y − x) dA, where R
is the region bounded by the lines y = 2x, y = 3x, y = x + 1, and y
= x + 2. Use the change of variables u = y x and v = y − x.

3) Four statistically independent random variables, X, Y, Z, W
have means of 2, -1, 1, -2 respectively, variances of X and Z are 9
and 25 respectively, mean-square values of Y and W are 5 and 20
respectively. Define random variable V as: V = 2X - Y + 3Z - 2W,
find the mean-square value of V (with minimum math).

1.) Consider two independent discrete random variables X and Y
with V(X)=2 and V(Y)=5. Find V(4X-8Y-9).
2.) Consider two independent discrete random variables X and Y
with SD(X)=16 and SD(Y)=9. Find SD(5X-2Y-13). (Round your answer to
1 place after the decimal point).
3.)Consider two discrete random variables X and Y with V(X)=81
and V(Y)=36 and correlation ρ=0.7. Find V(X-Y). (Round your answer
to 1 digit after the decimal point).

using the change of variable x =u/v, y=v evaluate "double
integral(x^2+2y^2)dxdy: R is the region in the first quadrant
bounded by the graphs of xy=1, xy=2, y=x, y=2x

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