Simply saying differential equation is any equation that contains derivatives. For example, `(y'')/t+y'+ty=0` , `(y''')^4+sqrt(y'')-y'=5t` are all differential equations.
There can be used different notation: either `y^((n))` or `(d^ny)/(dt^n)` .
So, 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`.
Differential equation is ordinary if it contains derivatives with respect to only one variable. Differential equation is called partial if it contains derivatives with respect to more than one variable. So, 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).
Order of differential equation is the order of the highest derivative in the equation. For example, `y''-ty=y'''` is the third-order, while `y'-ty=2` is first-order.
A solution of differential equation is unknown function (y or u) along with independent variables that satisfy 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 has a set of solutions; to restrict solution to one particular solution additional conditions are added. To find particular solution of n-th order differential equation, n conditions must be added.
Conditions are called initial if values of functions and its derivative are given at the same value of the independent variable and the corresponding problem is called initial value problem (IVP). For example, `y''+3y'+2y=0` y(1)=1, y'(1)=3 is IVP. And conditions y(1)=1, y'(1)=3 are initial. Another example of initial conditions for above problem is y(0)=0, y'(0)=3.
Conditions are called boundary they are given at more than one value of the independent variable and the corresponding problem is called boundary value problem (BVP). For example, `y''+3y'+2y=0` y(1)=1, y(2)=3 is IVP. And conditions y(1)=1, y(2)=3 are boundary. Another example of initial conditions for above problem is y(0)=0, y'(4)=5.
So, particular solution is solution that satisfies differential equation and given conditions. General solution is a solution in the most general form that only satisfies differential equation without satisfying any additional conditions.
Solution is called explicit if `y=y(t)` , in other words if the only appearance of y is on the left side and raised to first power. For example `y=e^t+ln(t)/t` and `y=ln(sqrt(t))+t` are explicit solutions.
Solution is called implicit if we can't find for y form `y=y(t)`. For example `y^3+ln(ty)=0` and `sqrt(ye^y)=t` are implicit solutions.
Before moving further it is essential before solving any differential equation ask yourself following 3 questions:
Does given differential equation has solution? Not all differential equations have solutions, for example `(y')^2=-1` doesn't have any solutions because left part is never negative and right part is negative.
If 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 differential equation has solution, is it possible to find it? Answer is not always yes.