The simplest differential equation (with positive order) is y′ = f(x). By the fundamental theorem of calculus, this has the solution y = c + ∫f, where c is a constant determined by some initial condition, such as y(0) = c.
In general, nth order differential equations tend to have a family of solutions, with n independent constants determined by initial conditions. To illustrate this, let's repeat the above procedure n times, for an nth order equation.
If y′′ = f(x), we can think of y′ as its own function z(x). We already know how to solve z′ = f(x). The result is c + ∫f. Replace z with y′ and solve the next differential equation.
y′ = c + ∫f
y = d + cx + ∫∫f
By induction, we can solve an nth order differential equation that sets the nth derivative of y equal to f(x). Integrate n times, and add in the expression c0+c1x+c2x2+…cn-1xn-1. These constants can usually be determined by initial conditions. If you are given y(0), y′(0), y′′(0), etc, assign these values to c0, c1, c2, etc, and you have the solution in hand.
y′′ = sin(x)
y = c0 + c1x - sin(x)
So far we have viewed y(x) as a real function of a real variable, however, x and y could be complex. Everything on this page remains valid, because complex integration is still the inverse of complex differentiation.