Find the region in the xy plane in which the equation [(x - y)^2 - 1] u_xx + 2u_xy + [(x - y)^2 - 1] u_yy = 0 is hyperbolic. The complete problem is in the attached file.

Linear PartialDifferentialEquation (II)
Non- Homogeneous Linear PartialDifferentialEquation with Constant Coefficients
Problem: Find the solution of the equation (D2 - D'2 + D -

Please help with the following problem. Provide step by step calculations for each problem.
Consider the Helmholtz partialdifferentialequation:
u subscript (xx) + u subscript (yy) +(k^2)(u) =0
Where u(x,y) is a function of two variables, and k is a positive constant.
a) By putting u(x,y)=f(x)g(y), derive ordinary diff

Hi,
Please help working on
section 1.1 problems 2,4,8,14,16
section 1.2 problems 6,10,20,24,27
thank you
See attached
Classify each as an ordinary differentialequation (ODE) or a partialdifferentialequation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ODE, ind

The heat transfer in a semi-infinite rod can be described by the following PARTIALdifferentialequation:
∂u/∂t = (c^2)∂^2u/∂x^2
where t is the time, x distance from the beginning of the rod and c is the material constant. Function
u(t,x) represents the temperature at the given time t and p

A) Solve the following differentialequation by as many different methods as you can.
(See attachment for equation)
b) There is a type of differentialequation which will always be solvable by two different methods. What type of differentialequation is it and which other method can always be used to solve it?
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1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function.
2) The wave equation (c^2 ∂^2 u / ∂ x^2 = ∂^2 u / ∂ t^2) and the heat equation (c ∂^2 u / ∂

Find the General Solution of the equations.
(a) r = a2t
(b) r - 3as + 2a2t = 0 where r = ∂2z/∂x2 , s = ∂2z/∂x∂y, t = ∂2z/∂y2
(c) (2D2 + 5DD′ + 2D′2)z = 0 (d) ∂3z/∂x3 - 3∂3z/∂x2∂y + 2∂3z/∂x∂y2 = 0

(1) Use Laplace Transforms to solve DifferentialEquation
y'' - 8y' + 20 y = t (e^t) , given that y(0) = 0 , y'(0) = 0
(2) Use Laplace Transforms to solve DifferentialEquation
y''' + 2y'' - y' - 2y = Sin 3t , given that y(0)=0 , y'(0)=0 ,y''(0)=0, y'''(0)=1
Note: To see the questions in their mathematic