Lecture 03 - Jan 8

Solve

1.

2.

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Problem 1

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Problem 2

IF:

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Homogeneous Equations

If is a homog. DE, then change of var. as transform the DE into separable eq. in the var. .

Proof

We have .

Given that it is a homog eq. and hence we have,

Integrate to get soln.

Problem

Solve,

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Initial and Boundary Value Problems

(I)

When are constants of is the initial point.

1. Existence
2. Uniqueness } well-posed problem.
3. Stability

Even if one not satisfied } ill-posed problem.

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Lemma

If is a linear func., then

where are constants of

Proof

Let be the basis of . Then, any can be written as,

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Theorem

If , is a homog. lin. eq of order defined on , then it can be written as

for each where are some func. defined on .