The solution to the differential equation subject to the given initial condition is log P = 0.07t + log(30)
Differential equation in which the variables can be separated from one from one another are called separable differential equations
A general form to write a separable differential equation is
dy/dx = f(x)g(y)
where the variable x and y can be separated from each other
According to the question,
We have to solve dP/dt=0.07P by using separation of variable
=> dP/dt=0.07P
Dividing P both sides and multiplying dt
=> dP / P = 0.07 dt
Now Integrating both the sides
=> [tex]\int\limits {\frac{1}{P} \, dP = \int\limits {0.07} \, dt[/tex]
=> log P = 0.07t + c ------(1)
It is given that P(0)=30
which means at t = 0 , the value of P = 30
Substituting t = 0 and P = 30
log(30) = 0 + c
=> c = log(30)
Substituting the value of c back to equation (1)
log P = 0.07t + log(30)
=> log(P / 30) = 0.07t is our solution
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