# The rate of change of a quantity T is given by \frac{dT}{dt} = 6e^{2t} , and T(0) = 10 ....

## Question:

The rate of change of a quantity {eq}T {/eq} is given by {eq}\frac{dT}{dt} = 6e^{2t} {/eq}, and {eq}T(0) = 10 {/eq}.

Find {eq}T(t) {/eq}

## Differential Equations:

We have been given a differential equation which has an exponential term. We will change the differential equation into a variable separable form. Then we need to find the constant.

## Answer and Explanation:

{eq}\frac{dT}{dt} = 6e^{2t} {/eq}

We will change the differential equation to variable separable form:

{eq}f(x)dx=g(y)dy\\ dT=6e^{2t}dt {/eq}

Now, all we need to do is apply integration:

{eq}\int dT=\int 6e^{2t}dt {/eq}

Applying the standard integral formula:

{eq}T=3e^{2t}+C {/eq}

Now we need to find the constant of integration which we will find by the given initial condition:

{eq}T(0)=10\\ 10=3+C\\ C=7\\ T=3e^{2t}+7 {/eq}

#### Learn more about this topic: Separable Differential Equation: Definition & Examples

from GRE Math: Study Guide & Test Prep

Chapter 16 / Lesson 1
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