A differential equation is any equation which contains an unknown function and one of its unknown derivatives. If only ordinary derivatives are present, then the equation is called an ordinary differential equation and if partial derivatives are involved, then the equation is called a partial differential equation. D.E play an extremely important role in applied math, engineering and physics.
If y= f(x) is an unknown function, an equation which involves at least one derivatives of y w.r.t x is called an ordinary differential equation. The order of the D.E is the order of the highest derivative present in the equation and the degree of the D.E is the degree of the highest order derivative after clearing the fraction powers.
1. dy/dx=2x in this the Order =1, Degree =1
The first order and first degree equation will be in the form
dy/dx = f(x,y)
In homogeneous equation
First-order linear constant coefficient ordinary differential equation:
du/dx = cu + x2
Second-order linear ordinary differential equation:
d2u/dx2 - x du/dx + u = 0
Second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator:
d2u/dx2 + w2u = 0.
First-order nonlinear ordinary differential equation:
du/dx = u2 + 4.
Second-order nonlinear (due to sine function) ordinary differential equation describing the motion of a pendulum of length L:
L d2u/dx2 + g sin u = 0.
Homogeneous first-order linear partial differential equation:
Homogeneous second-order linear constant coefficient partial differential equation of elliptic type, the Laplace equation:
Third-order nonlinear partial differential equation, the Korteweg–de Vries equation: