Introduction to Differential Equations

To put it simply, a differential equation is any equation that contains derivatives. For example, `(y'')/t+y'+ty=0` and `(y''')^4+sqrt(y'')-y'=5t` are both differential equations.

Different notations can be used: either `y^((n))` or `(d^ny)/(dt^n)`.

So, the above equations can be rewritten with another notation as follows: `1/t (d^2y)/(dt^2)+(dy)/(dt)+ty=0` and `((d^3y)/(dt^3))^4+sqrt((d^2y)/(dt^2))-(dy)/(dt)=5t`.

A differential equation is ordinary, if it contains derivatives with respect to only one variable. A differential equation is called partial, if it contains derivatives with respect to more than one variable. So, the above equations are examples of ordinar differential equations (ODE), and both `((partial u)/(partial t))^2+(partial u)/(partial x)=x+t` and `(partial u)/(partial t)+(partial u)/(partial x)=((partial u)/(partial v))^2` are examples of partial differential equations (PDE).

The order of a differential equation is the order of the highest derivative in the equation. For example, `y''-ty=y'''` is a third-order differential equation, while `y'-ty=2` is a first-order one.

A solution of a differential equation is an unknown function (`y` or `u`) along with the independent variables that satisfy the given differential equation.

For example, `y(t)=e^t` is a solution to the differential equation `y'-y=0`, because `y'=e^t` and `e^t-e^t=0`.

In general, each differential equation has a set of solutions; to restrict it to one particular solution, supplementary conditions are added. To find a particular solution of an n-th order differential equation, n conditions have to be added.

Conditions are called initial, if the values of a function and its derivatives are given at the same value of the independent variable, and the corresponding problem is called the initial value problem (IVP). For example, `y''+3y'+2y=0`, `y(1)=1`, `y'(1)=3` is an IVP, and the conditions `y(1)=1`, `y'(1)=3` are initial. Another example of initial conditions for the above problem is `y(0)=0`, `y'(0)=3`.

Conditions are called boundary, if they are given at more than one value of the independent variable, and the corresponding problem is called the boundary value problem (BVP). For example, `y''+3y'+2y=0`, `y(1)=1`, `y(2)=3` is a BVP, and the conditions `y(1)=1`, `y(2)=3` are boundary. Another example of boundary conditions for the above problem is `y(0)=0`, `y(4)=5`.

So, a particular solution is a solution that satisfies the differential equation and the given conditions. A general solution is a solution in the most general form that only satisfies the differential equation without satisfying any additional conditions.

A solution is called explicit, if `y=y(t)`, or, in other words, if the left-hand side equals `y` and the right-hand side doesn't contain `y`. For example, `y=e^t+ln(t)/t` and `y=ln(sqrt(t))+t` are explicit solutions.

A solution is called implicit, if we can't express `y` in terms of independent variables. For example, `y^3+ln(ty)=0` and `sqrt(ye^y)=t` are implicit solutions.

Before solving a differential equation, it is essential to ask yourself the following 3 questions:

  • Does the given differential equation has a solution? Not all differential equations have solutions, for example `(y')^2=-1` doesn't have any solutions, because the left side is never negative and the right side is negative.
  • If the differential equation has solutions, how many solutions are there? For example, `y'=sqrt(y)` has at least 2 solutions: `y=1/4 t^(2)` and `y=0`.
  • If the differential equation has a solution, is it possible to find it? The answer is not always yes.