 Hello, and welcome to this screencast. In this video, we will solve an example of a separable differential equation. Our example of a differential equation is dy dt is equal to 3y plus 6. To solve this differential equation, that means we want to find a function y of t, so that when we differentiate that function y with respect to t, our result is 3 times that original function y plus 6. So our first step to solving this equation is factoring out the 3 on the right side, and so we get 3 times the quantity y plus 2. Next, we divide both sides by y plus 2, and so on the right side we're left with just 3. Our next step is to integrate both sides with respect to t. And on the left side, we see that we can simplify the integral by cancelling the dt, and so we're left with the integral of 1 divided by y plus 2 dy. When we evaluate that integral on the left side, we get the natural log of the absolute value of y plus 2. And on the right side, we get 3t plus some unknown constant c1. I called this constant c1 because I know we're going to modify it as we solve this before we're done with our problem, and so we'll change the name of an unknown constant as we go along. To solve this for y, we know we need to eliminate the natural log on the left side of the equation, so we're going to exponentiate both sides. On the left side, we raise e to the power of the natural log of y plus 2, and on the right side, we raise e to the power of 3t plus c1. e to the x in natural log of x are inverse functions, so the left side simplifies to just y plus 2. On the right side, we can use properties of logs to rewrite that as e to the 3t times e to a power of c1. Now the second part of that, e to the c1, is e raised to the power of a constant, and even though we don't know the value of c1, we do know that e to the c1 is just a constant, and so we can replace that with just c, so e to the 3t times c. We're almost there, now we just need to subtract 2 from both sides of our equation, and the result is y is equal to c times e to the 3t minus 2, where c is an unknown constant. This means there are infinitely many solutions to our differential equation that differ by the choice of that constant c, just like there are infinitely many anti-derivatives of a function. But we're not quite done yet, we still want to check our solution. So to do this, we start with the left side of the equation, and we differentiate our function y with respect to t. And the result is c times e to the 3t, and then since we're using the chain rule, we multiply that by 3, the derivative of the exponent. So that's the left side of our differential equation using our solution for y. Next, on the right side of our differential equation is 3y plus 6, and we want to find this for our value of y. So we substitute in c e to the 3t minus 2 for y, and then next we can distribute the 3, so we get 3c e to the 3t minus 6 and plus 6. So that minus 6 and plus 6 simplifies to 0, and leaving us with 3c e to the 3t. So that's our result for the right side of our equation using our solution for y. We compare these two results, and we see that using our solution for y, the left side and the right side of the differential equation are equal. And so that means our solution y is equal to c times e to the 3t minus 2 is in fact the correct one. Thanks for watching.