Differential Equations Answers

Form Partial differential eq. of 2u=x^2/a^2 +y^2/b^2

Form Partial differential eq. of u=ax+(1-a)y+b

(1) obtain the solution for the initial value problem dy/dx+(cotx)y=xcscx,y(pi/2)=1

(2)solve the differential equation dy/dx=x(1+y^2)

(3) solve the differential equation dy/dx+(cotx)y=xcscx

the differential equation y"'-y''=8x^2. I know after solving the homogeneous part of the equation you get that 1 and 0 are roots of the equation. I know for the 8x^2 you're supposed to substitute the constant for a variable then that would be Ax^2 and that i have to derive three times, My question is, after i derive three times what would i do with the second and the first derivative? there isn't any y'' or y' where to replace that in the original equation. Thanks you very much!

can help me

Show that the function
i) 3 u(x, t) = A(x + ct)³ is a solution of the one-dimensional wave equation
ii) u( x, t) =exp(-ut) sin x , is a solution of the one-dimensional heat equation

The electrostatic potential V(x,y) in a rectangular region is governed by the two dimensional
Laplace equation:

∂2 V(x,y)/∂x2 + ∂2 V(x,y)/∂y2 = 0

determine V(x,y) in a rectangular region 0 ≤ x ≤ 20 and 0 ≤ y ≤ 40 for the following
boundary conditions:
V (0, y) =V(20, y) =V(x,0) = 0
and
V (x,40) =110V

A chain hangs over a nail with 2.0 m on one side and 6.0 m on the other side. If the
force of friction is equal to the weight of 1.0 m of the chain, calculate the time required
for the chain to slide off the nail.

Obtain the Fourier series for the following periodic function which has a period of 2π:

f(x) = x^2 for -π ≤x ≤π

A box is to have square base an open top and volume of 32 meter cube.find the dimension of the box that box that use the last amount of material.