 Hello, and welcome to this screencast. In this video, we will solve another example of a separable differential equation. The differential equation we will solve is dy dx is equal to y squared times cosine of x. Solving this differential equation means finding a function y of x such that when we differentiate that function y, the result is our original function y squared and then multiplied by cosine of x. So let's start just by rewriting our differential equation below. And next, we're going to separate our variables by dividing both sides by y squared. So y squared is in the denominator on the left side and on the right side we just have cosine of x. Now we want to integrate both sides of our equation with respect to x. On the left side, we notice that we can simplify by cancelling the dx, so on that side we are now integrating with respect to the variable y. For the integral on the left, we can use the power rule to integrate 1 over y squared. You might find it helpful to think of the integrand as y to the negative 2 power if you're using the power rule. And the result is negative 1 over y. On the right side, the antiderivative of cosine of x is sine of x. So we will write sine of x plus c1. I chose to write the constant as c1 instead of just c because I know we will make a change to it before we're done with our solution. Next, we multiply both sides by negative 1 and we get 1 over y is equal to negative sine of x. Instead of writing minus c1 here, I'm going to replace the unknown constant negative c1 with the simpler plus c, which still represents an unknown constant. Our last step for solving for y is taking the reciprocal of both sides. On the left side, this just means we'll have y. On the right side, we get 1 over negative sine of x plus c. Notice that that plus c is also in the denominator because we want the reciprocal of the entire expression negative sine of x plus c. And so that's our general solution to the differential equation. So we have y equals 1 over negative sine of x plus c. We note that there are really infinitely many different solutions which are determined by the specific value of that constant c. But before we finish, we want to check our solution just to make sure it really does satisfy that original differential equation. First, we'll consider the left side of the differential equation and that says dy dx. So dy dx means differentiating our function y with respect to x. And to make this step easier, we're going to rewrite y as the quantity negative sine of x plus c to a power of negative 1. To differentiate this, we're going to use the chain rule. And so we get negative 1 times the quantity negative sine of x plus c raised to a power of negative 2. And then we multiply by negative cosine of x, the derivative of the inside function. Since we're multiplying these two, we see that these two negative sines here are going to simplify to just a positive 1. And so we can rewrite this in a fraction with cosine of x in the numerator and negative sine of x plus c squared in the denominator. So that's the left side of our differential equation. Now we're going to consider the right side of our differential equation using our solution for y. To evaluate y squared cosine of x, we simply substitute our solution for y and square it. And then we multiply by cosine of x. When we rewrite this fraction, we see that it is the same as our result for the left side that we found above. So we have that fraction and it's the same as that fraction we found above for the left side. And that shows us that the solution we found for y really is valid because it does make the differential equation true because the left and right side of the equation are equal. And that ends our screencast. Thanks for watching.