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I assume you mean
ANSWER
An expression for P(t) is
EXPLANATION
This is a separable differential equation. Treat M and k as constants. Then we can divide both sides by M - P to get the P term with the differential dP and multiply both sides by dt to separate dt from the P terms
Integrate both sides of the equation.
Note that for the left-hand side, u-substitution gives us
hence why
Now we use the definition of the logarithm to convert into exponential form.
The definition is
so applying it here, we get
Exponent properties can be used to address the constant C. We use
here:
If we assume that P(0) = 0, then set t = 0 and P = 0
Substituting into our original equation, we get our final answer of
ANSWER
An expression for P(t) is
EXPLANATION
This is a separable differential equation. Treat M and k as constants. Then we can divide both sides by M - P to get the P term with the differential dP and multiply both sides by dt to separate dt from the P terms
Integrate both sides of the equation.
Note that for the left-hand side, u-substitution gives us
hence why
Now we use the definition of the logarithm to convert into exponential form.
The definition is
so applying it here, we get
Exponent properties can be used to address the constant C. We use
If we assume that P(0) = 0, then set t = 0 and P = 0
Substituting into our original equation, we get our final answer of