# 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.