manipulating

1. manipulating differential operator

How does the top of part B become the bottom of part B?
2. Manipulating Differentials: Method of Characteristics

I've been looking over some papers on the Method of Characteristics but I've found something I'm not sure about. For quasi-linear equations of the form $$a(x,y,u)u_x + b(x,y,u)u_y = c(x,y,u)$$ I have found two methods. One involves writing \frac{\mathrm dx}{a(x,y,u)} = \frac{\mathrm...
3. Manipulating differential equation

v(x)p(x) = d/dx (D(x)p(x)) can be rearranged to 1/(D(x)p d/dx(D(x)p(x)) = d/dx ln(D(x)p(x)) where d/dx are partial differentials. Can anyone explain how this works to me? Thanks
4. Manipulating limits

Hallo! I need some help. lim x->4 ((2x^3-128)/(sqrt(x)-2)) i have tried manipulating the equation with conjugates but i just keep getting nowhere. wolfram has the limit at 384 please help!
5. Help with algebra for a proof: manipulating an expression

I have a proof that I have to do. for all positive natural numbers n and for any integers a and b such that a != b a^n - b^n is divisible by a-b the induction hypothesis is to assume the a^k - b^k is divisible by a-b I need to show that a^(k+1) - b^(k+1) is divisible by (a-b). I have seen the...