Differential equations - Mu''(t) + ku(t) = 0

Solve using integrating factor and Bernoullis.

The question is attached along with relevant course notes. Only part ii needs to be answered. Not everything with indices is a tensor. If theta and A^u transform as a scalar and a contravariant vector under general coordinate transformations, respectively, then show that: (See attached formula) Is a tensor, but that:

Differential Equation (IX): Formation of Differential Equations by Elimination Eliminate the arbitrary constants from the equation: y = Ae^x + Be^2x + Ce^3x. Make sure to show all of the steps which are involved.

I am asking for step-by-step workings for all attached problems.

Question 1. 1) Find a vector normal to the surface z + 2xy = x2 + y2 at the point (1,1,0). 2) Determine if there are separable differential equations among the following ones and explain: a) dy/dx=sin(xy), b) dy/dx = (xy)/(X+y) c) dr/d(theta) = (r^2+1)cos(theta) 3) Find the general solution of the differential

For each of the following ordinairy differential equations, indicate its order, whether it is linear or nonlinear, and whether it is autonomous or non-autonomous. a) df/dx +f^2=0 (See attachment for all questions)

A) Solve the following differential equation by as many different methods as you can. (See attachment for equation) b) There is a type of differential equation which will always be solvable by two different methods. What type of differential equation is it and which other method can always be used to solve it? ---

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.

Please follow all instructions and answer accordingly. See the attached file. Consider a projectile of mass m