Introduced in Semester 1.
Notations
When the case of 2 independent variables is considered, x and ysareassumedtobetheindependentvariablesandz$
is assumed be the dependent variable. The following notations are used in this section.
p=∂x∂z;q=∂y∂z;r=∂x2∂2z;s=∂x∂y∂2z;t=∂y2∂2z;
If there are n independent variables, they are considered as x1,x2,…,xn. In this case the following notations are used:
pi=∂xi∂z;i=1,2,…,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, ϕ(u,v)=0 is an arbitrary function. And
u(x,y,z)=c1 and v(x,y,z)=c2 are two independent solutions of:
P(x,y,z)(dx)=Q(x,y,z)(dy)=R(x,y,z)(dz)
Here c1 and c2 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.