Partial Differential Equations

Introduced in Semester 1.

Notations

When the case of 2 independent variables is considered, $x$ and $ys are assumed to be the independent variables and$z$ is assumed be the dependent variable. The following notations are used in this section.

$$p = \frac{\partial z}{\partial x} ; \;\; q = \frac{\partial z}{\partial y} ; \;\; r = \frac{\partial^2 z}{\partial x^2} ; \;\; s = \frac{\partial^2 z}{\partial x \partial y} ; \;\; t = \frac{\partial^2 z}{\partial y^2} ; \;\;$$

If there are $n$ independent variables, they are considered as $x_1, x_2, \dots, x_n$. In this case the following notations are used:

$$p_i = \frac{\partial z}{\partial x_i} \;\;;\;\;i=1,2,\dots,n$$

Types

Suppose $f(x,y,z,p,q) = 0$. $$

Linear

When it is linear in $p,q,z$. When it is of form: $$

$$P(x,y)p + Q(x,y)q = R(x,y)z + S(x,y)$$

Semi-linear

When it is linear in $p,q$ and $p,q$ are functions of $x,y$ only. When it is of form:

$$P(x,y)p + Q(x,y)q = R(x,y,z)$$

Quasi-linear

When it is linear in $p$ and $q$. Whe it is of form:

$$P(x,y,z)p + Q(x,y,z)q = R(x,y,z)$$

Non-linear

When it does not satisfy any of the above type.

Lagrange’s Equation

Quasi-linear partial differential equation of order one.

Solving method

General solution of Lagrange’s equation, $\phi(u,v)=0$ is an arbitrary function. And $u(x,y,z) = c_1$ and $v(x,y,z)=c_2$ are two independent solutions of:

$$\frac{(dx)}{P(x,y,z)} = \frac{(dy)}{Q(x,y,z)} = \frac{(dz)}{R(x,y,z)}$$

Here $c_1$ and $c_2$ are arbitrary constants. At least one of $u$ or $v$ must contain $z$. $u$ and $v$ are independent if $u=v$ is not a constant.