The first definition which we must cover is that of differential equation. A differential equation is any equation that comprises derivatives, either partial derivatives or ordinary derivatives.

There is one differential equation which everybody probably identifies, i.e. Newton's Second Law of Motion. If an object of mass m is moving along with acceleration a and being reacted on force F so by Newton's Second Law we get:

F = ma   ...............(1)

To notice that it is actually a differential equation we require to rewrite it a little. Firstly, keep in mind that we can rewrite the acceleration, a, in one of two methods.

a = dv/dt          or         a = d²u/dt²     ...........................(2)

Here v = the velocity of the object and

u = the position function of the object at any time t.

We should also keep in mind at such point where the force, F may also be a function of time, position and/or velocity.

Thus, with keeping all these things in mind Newton's Second Law can now be written as a differential equation in terms of either the velocity v, or the position u, of the given object as given below:

m(dv/dt) = F (t,v)     ........................(3)

m(d²u/dt²) = F (t,u,du/dt)     ........................(4)

Hence, here is our first differential equation. We will consider both forms of this later.

There are several more illustrations of differential equations.

ayⁿ +by' + cy = g (t)   ..........................(5)

sin(y) d²y/dx² = (1 - y)(dy/dx) + y² e^-5y   .........................(6)

y⁽⁴⁾ + 10 y^m - 4y' + 2y = cos (t)      ......................(7)

α² (∂²u/∂x²) = ∂u/∂t        ........................(8)

α² u_xx = u_tt     ..........................(9)

(∂³u/∂²x ∂t) = 1 + ∂u/∂y    ...................(10)