How to differentiate between, Partial and ordinary differential equations, Linear and non linear differential equations, Order and degree of a differential equation.

Answer 1

Partial and ordinary differential equations

An ordinary differential equation (ODE) is an equation that involves ordinary derivatives with respect to a single independent variable.

Example 1

d²x/dt² + a(dx/dt) + kx = 0

where t is the independent variable and x is the dependent variable while a and k are coefficients.

Partial differential equation (PDE) is an equation that involves partial derivatives with respect to more than one independent variables.

Example 2

∂u/∂x - ∂u/∂y = x - 2y

where x and y are independent variables and u is the dependent variable.

Order of a differential equation

The order of a differential equation is the order of the highest-order derivative present in the equation. For example,

from example 1

d²x/dt² is the highest order derivative in the equation, which is raised to the second power. Making it a second order differential equation.

from example 2

Only first order partial derivatives occur in the equation making it a first order differential equation.

Degree of a differential equation

This can be defined as the degree of the highest ordered derivative which occur in the equation.

So, example 1 is of degree (1), example 2 is of degree (1) also.

Linear and non linear differential equations

A linear differential equation is an equation which the dependent variable y and it’s derivatives appear in additive combinations of their power. Non-linear differential equations do not follow this pattern. For instance

(1 - x)y’’ + 4xy’ + y = cosx

Or

d²u/dr² + du/dr + u = cos(r + u)

The above are linear equations. Do you see the pattern? from y’’ to y’ to y

Now take a look at these

x(d³y/dx³) + y(d⁴y/dx⁴) + y = sinx

Or

d²y/dx² + y³ = 0

The above are examples of non-linear differential equations because y and it’s derivatives don’t appear in additive combinations of their power.