Question

Use separations of variables to solve the Differential Equation:

K dN/dt = -r(N-K)(N-A)

After doing partial fractions and the integration I get:

(N-A)/(N-K) = C_1 e^((A-K)(-rt)/K)

I am stuck solving for N

Answer 1

\[\frac{ N-A }{N-K }=C_1e^{\frac{ (A-K)(-rt) }{ K }}\]
I am stuck in this part, solving for N

Answer 2

\[\large\rm \frac{ N-A }{N-K }=c_1e^{stuff}\]Multiply both sides by (N-K),\[\large\rm N-A=(N-K)c_1 e^{stuff}\]

Answer 3

Distribute,\[\large\rm N-A=N c_1 e^{stuff}-K c_1 e^{stuff}\]Let's subtract N to move it to the right side,
and add Kc_1 e^(stuff) to the other side by adding,\[\large\rm K c_1 e^{stuff}-A=N c_1 e^{stuff}-N\]Then factor an N out of each term on the right side,\[\large\rm K c_1 e^{stuff}-A=N (c_1 e^{stuff}-1)\]And divide to isolate your N,\[\large\rm \frac{K c_1 e^{stuff}-A}{c_1 e^{stuff}-1}=N\]

Answer 4

If your K is just a constant,
you can probably just absorb it into the c.\[\large\rm \frac{c_2 e^{stuff}-A}{c_1 e^{stuff}-1}=N\]

Answer 5

Thanks for the help.

Answer 6

:D