# Find the solution to the differential equation: \frac{dy}{dt}=0.6(y-200), if y=75 when t=0

## Question:

Find the solution to the differential equation: {eq}\frac{dy}{dt}=0.6(y-200){/eq}, if {eq}y=75{/eq} when {eq}t=0{/eq}

## Differential Equations:

We have a differential equation which has a constant term and a linear term. There are many types of differential equations as a linear differential equation and we have a different way to solve it but here we will apply variable separable form.

{eq}\frac{dy}{dt}=0.6(y-200) {/eq}

We will change the differential equation to variable separable form:

{eq}f(x)dx=g(y)dy\\ \frac{dy}{y-200}=0.6 dt {/eq}

Now we will integrate on both sides:

{eq}\int \frac{dy}{y-200}=\int 0.6 dt {/eq}

We will apply the standard integral formulas:

{eq}\ln (y-200)=0.6 t+C {/eq}

Now we have to find the value of constant of integration by the given initial condition:

{eq}y(0)=75\\ C=\ln (-125)\\ \ln (y-200)=0.6 t+\ln (-125) {/eq}

is the solution.