Question

I'm looking at some of the examples in a Wikibooks book on Ordinary Differential Equations, and I can't figure out part of one. Link below and specified example.

Answer 1

ok

Answer 2

http://en.wikibooks.org/wiki/Calculus/Ordinary_differential_equations#Variation_of_constants

In the first example for Variations of Constants, I don't understand how, for y'', whoever wrote the entry got the second derivative to have the term 2x dz/dx; I got 4x dz/dx.

Answer 3

\[y = zx^2\]\[y' = 2xz \frac {dx}{dx} + x^2 \frac {dz}{dx}\]\[y'' = [(2x \frac {dz}{dx} + 2z \frac {dx}{dx}) + (2z \frac {dz}{dx} \frac {dx}{dx} + \frac {d^2 z}{dx^2}x^2)]\]

Answer 4

Appropriately eliminating dx/dx, simplifying, etc, I get:

Answer 5

\[x^2 \frac {d^2 z}{dx^2} + 4x \frac {dz}{dx} + 2z = y''\]

Answer 6

why 4?

Answer 7

y''. Right up there, eliminating dx/dx, 2x dz/dx + 2x dz/dx = 4x dz/dx.

Answer 8

Then I still have an extra term that isn't in the solution, if I'm seeing straight.