Module 1

Important

Partial differential equations,
- Formation of partial differential equations
  - elimination of arbitrary constants
  - elimination of arbitrary functions,
- Solutions of a partial differential equations,
  - Equations solvable by direct integration
- Linear equations of the first order Lagrange’s linear equation,
- Non-linear equations of the first order - Charpit’s method,
- Solution of equation by method of separation of variables.

Partial Differential Equations

Sample Problems

Solve (3 Marks)

\frac{\partial^{2} z}{\partial_{x} \partial_{y}} = x^{2} y

Derive a partial differential equation from the relation (3 Marks)

z = f (x + a t) + g (x - a t)

Solve (7 Marks)

x (y - z) p + y (z - x) q = z (x - y)

Use Charpit's method to solve

q + x p = p^{2}

Find the differential equation of all spheres of fixed radius having their centers in the $x y$ -plane.
Using the method of separation of variables , solve

\partial \frac{u}{\partial x} = 2 \frac{\partial u}{\partial t} + u

where $u (x, 0) = 6 e^{- 3 x}$