7.1 Separable Differential Equations
One technique to solve differential equations is the method of separation of variables. Let’s look at an example:
What is the general solution to dydx=yx2? To solve this expression we collect the variables involving x and one side of the equation, and the variables involving y on the other:
1ydy=x2dx.
Now the next step is to determine the antiderivative of both sides of expression:
∫1ydy=ln(y)+C.∫x2dx=13x3+C.
Finally since both sides are equal we can solve for the dependent variable y. One thing to note: usually for antiderivatives we always include a +C. For solving differential equations it is okay just to keep only one +C, which usually is best on the side of the independent variable:
ln(y)=13x3+C→eln(y)=e13x3+C→y=Ce13x3.
We are in business! So here is a general technique approach to solving a differential equation via separation of variables:
- Separate the variables on one side of the equation.
- Integrate both sides individually.
- Solve for the dependent variable.
If we solve this equation using separation of variables we havUsing your work above as a guide, solve this differential equation to determine a solution y(x).