Solving Anti-Derivative Equations

This form of a differential equation basically states that one function is the derivative of another. For instance, if $y(x) = 2x$, and $y'(x) = f(x)$, then $f(x) = 2$. The following differential equation is true:

$(2x)' = 2$

If this notation is confusing, recall that it is equivalent to saying

$\frac{d}{dx}(2x) = 2$

For differential equations of this form $y'(x) = f(x)$, one can solve for $y$ simply by integrating both sides of the differential equation. Do note, however, that a $+C$ will be tacked onto the general solution of the differential equation since the general solving of differential equations uses definite integrals.

At the bottom, I will attach a worksheet with some practice on separable differential equations, which is basically another name for what you just learned from the explanation. When you are given the form $\frac{dy}{dx} = f(x)$, you just need to multiply both sides by $dx$ in order to get the equation into a form you can just integrate. If there is a $y$ with the $x$, divide by the part containing the $y$ so that you have a function of $y$ times $dy$ equal to $f(x)dx$. Then integrate and you are done!

So the worksheet is attached here. Best of luck, and don't forget to check your answers at the bottom when you're done!

Perhaps the simplest differential equation is the anti-derivative equation, $y'(x) = f(x)$, where $f$ is a given function of $x$. By the FTC, the general solution is $y(x) = C + \int_0^x f(t)\,dt$, where $C = y(0)$, the constant of integration, is determined by the initial conditions (or sometimes other data).