dP/dt = k(P-a)
dP/dt = kP-ka
dP/P = k-ka
is this right so far? was i supposed to distribute??? what do i need to do next???
The third equation does not follow from the second.
This looks like a differential equation they want you to solve.
What you need to do is integrate both sides of
dP/(P-a) = k dt
ln (P-a) = kt + constant
C e^(kt) = P-a
P = a + C e^(kt)
C is an arbitrary constant that you will have to determine from an initial condition.
check: dP/dt = k C e^(kt) = k (P-a)
Yes, you're on the right track so far! To simplify further, you need to integrate both sides of the equation. Here's the step-by-step process:
1. Start with the equation: dP/dt = k(P-a)
2. Separate the variables by multiplying both sides of the equation by dt: dP = k(P-a) dt
3. Now, divide both sides of the equation by (P-a) to isolate the P variable: dP / (P-a) = k dt
4. Integrate both sides of the equation with respect to t. The integral of 1/t dt is ln|t|, so the left side becomes ln|P-a| and the right side becomes kt + C, where C is the constant of integration.
5. Now, we have ln|P-a| = kt + C
6. To solve for P, we can exponentiate both sides of the equation. Remember that e^ln|P-a| = |P-a|. Therefore, the equation becomes |P-a| = e^(kt + C)
7. Since we are working with a differential equation, we usually express the solution in terms of the absolute value |P-a| = Ce^(kt), where C is a positive constant.
8. Finally, we can solve for P by removing the absolute value: P-a = Ce^(kt)
9. Rearrange the equation to solve for P: P = Ce^(kt) + a
Now, you have the solution to the differential equation dP/dt = k(P-a) in terms of the constants C, k, and a.