Method: Separable Differential Equations

Step 1: Identify your two variables.

Step 2: Separate your equation by variables on to either side of the equals.

This will mean treating your differential notation , $\frac{\mathit{dy}}{\mathit{dx}}$, like two variables being divided. You will would want the $\mathit{dx}$ piece to be on the side with the x’s , or the $\mathit{dy}$ piece to be on the side with the y’s , or the $\mathit{dt}$ to be on the side with the t’s . Make sure that you are always multiplying or dividing variables by the $\mathit{dx},\mathit{ dy},\mathit{ dt}$ piece. You are about to take an antiderivative, and an antiderivative is normally of the form $\int \mathit{equation\; dx}$. You are never adding or subtracting your $\mathit{dx},\mathit{ dy},\mathit{ dt}$ piece.

Step 3: Take the antiderivative of both sides of the equation.

This antiderivative process will be the same as a standard indefinite integral process. You will only need a single +C value that will work for both antiderivatives. It is best to put this +C on the independent variable side ( x -variable) of the equation. Although it does not matter at this step of the process which side you choose.

Step 4: Solve the equation you found in Step 3 to get the dependent variable ( y -variable) alone on one side of the equals.

Step 5 (If asked to find a particular solution): Sometimes you will be given an actual value or point that the solution needs to go through (i.e., an $(x,y)$). You will use that value to find the +C value for your specific situation. This is just like the method you would apply in an initial value problem you learned earlier.