Partial Differential Equations

Partial differential equations (PDE’s) are equations that involve rates of change with respect to continuous variables. In other words, it is a mathematical equation that involves two or more independent variables, an unknown function (dependent on those variables), and partial derivatives of the unknown function with respect to the independent variables. 

A partial differential equation (PDE) for the function u(x1,… xn) is an equation of the form:

A simple PDF is written as:

\(\frac{\partial u}{\partial x}(x, y) = 0\)

 

\(f(x_{1}, ….x_{n};u, \frac{\partial u}{\partial x_{1}}, ….\frac{\partial u}{\partial x_{n}};\frac{\partial^2 u}{\partial x_{1}\partial x_{1}},…..\frac{\partial^2 u}{\partial x_{1}\partial x_{n}};….) = 0\)

 

This relation implies that the function u(x,y) is independent of x. However, the equation gives no information on the function’s dependence on the variable y. Hence the general solution of this equation is u(x, y) = f(y)

where f is an arbitrary function of y. The analogous ordinary differential equation is

\(\frac{\partial u}{\partial x}(x)= 0\)

 

which has the solution u(x) = c, where c is a constant value.

Solved Example

Question: Find the partial derivative of x3+ y3 – 3xy with respect to x.

Solution:

\(\frac{\partial }{\partial x}(x^{3}+y^{3}- 3xy)\)

 

\(=\frac{\partial }{\partial x}x^{3}+ \frac{\partial }{\partial x}y^{3}- \frac{\partial }{\partial x}3xy\)

 

\(=3x^{2}+0-3y\)

 

\(=3x^{2}-3y\)

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