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Sahithyan's S2
Sahithyan's S2 — Methods of Mathematics

Partial Differential Equations

Introduced in Semester 1.

Notations

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

p=zx;    q=zy;    r=2zx2;    s=2zxy;    t=2zy2;    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 nn independent variables, they are considered as x1,x2,,xnx_1, x_2, \dots, x_n. In this case the following notations are used:

pi=zxi    ;    i=1,2,,np_i = \frac{\partial z}{\partial x_i} \;\;;\;\;i=1,2,\dots,n

Types

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

Linear

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

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

Semi-linear

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

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

Quasi-linear

When it is linear in pp and qq. Whe it is of form:

P(x,y,z)p+Q(x,y,z)q=R(x,y,z)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, ϕ(u,v)=0\phi(u,v)=0 is an arbitrary function. And u(x,y,z)=c1u(x,y,z) = c_1 and v(x,y,z)=c2v(x,y,z)=c_2 are two independent solutions of:

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

Here c1c_1 and c2c_2 are arbitrary constants. At least one of uu or vv must contain zz. uu and vv are independent if u=vu=v is not a constant.