Prove Integral C: Step-by-Step Guide to Homework Statement

[moses474]

Homework Statement

prove,
integral C = b ln (y ) – py – rx + d ln ( x )
How do i do this? can someone help me out step by step? Thanks!

Homework Equations

f = b - py
g = rx - d

dx/dt = x (b - py)
dy/dt = y (rx -d)

The Attempt at a Solution

Well I was given those equations which my group has used to model a predator prey model. My group member "got" the integral some how but now we need to do a proof and are at a loss, any help would be great!

 

[lippyka]

Let's start by rewriting the equation in terms of derivatives: Integral C = b ln (y ) – py – rx + d ln ( x ) Integral C = b ln (dy/dt) – py – rx + d ln ( dx/dt )Now, let's substitute the equations for dx/dt and dy/dt into the integral: Integral C = b ln (y (rx -d)) – py – rx + d ln ( x (b - py))Let's divide each side by -d: (-d) Integral C = -d b ln (y (rx -d)) + dpy + drx - d d ln ( x (b - py))Let's factor out the common terms: (-d) Integral C = -d [b ln (y (rx -d)) + py - d ln ( x (b - py))] + drx Let's use the fact that ln (ab) = ln(a) + ln(b) to simplify the equation: (-d) Integral C = -d [b ln(y) + b ln(rx -d) + py - d ln(x) - d ln(b - py)] + drx Let's group -d and b together: (-d) Integral C = -d [b (ln(y) + ln(rx -d)) + py - d (ln(x) - ln(b - py))] + drx Let's move the terms outside of the brackets to the left side of the equation: Integral C = b ln (y ) – py – rx + d ln ( x ) + drx -d[b (ln(y) + ln(rx -d)) + py - d (ln(x) - ln(b - py))]The left side of the equation is the same as the original equation so we have proven that the integral is equal to the original equation.