Differential Equations of the form $\frac{dy}{dx} = f(x)$

The first differential equation that we're going to explore is something we have already seen. A differential equation is any equation that involves a derivative, and we've been looking at equations like this since we first started doing integral calculus. For example, we could say that

$$\frac{dy}{dx} = x^2$$

is a differential equation. We would specifically call it a first-order differential equation because we only have the term $\frac{dy}{dx}$ in it, and no higher-order terms like $\frac{d^2y}{dx^2}$.

We also know how to solve differential equations in this form, since we can integrate both sides of the equation to reverse the dervative that has been applied to $y$, and we would find that this differential equation is satisfied by the family of equations

$$y = \frac{x^3}3 + C, \quad C \in \R$$

At this stage, we don't have any information that tells us whether any values of $C$ are invalid, so we must specify that $C$ could take any real value. We would then say that this family of equations, parameterised by $C$, is the solution to the differential equation $\frac{dy}{dx} = x^2$.

...

Log in or sign up to see more