Lecture 02 - Jan 7

Previously Learnt

• Math modelling
• Formulas for DE
• Solns for DE

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Soln. contains arbitrary constants.

• → General soln. of DE.
• → When assign particular values for arb const. we get particular soln.

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Classification of DE

• Variable separation
• Exact equation
• Homogeneous equation
• Any other reduces to the above.

Separable Equation

Soln is obtained by integration.

Integrate to get soln.

Problems

1.

2.

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Exact DE

This is said to be an exact DE if a func. s.t.

So, ,

So if DE is exact, .

Converse:

→ Suppose the DE satisfies, .

Should show DE is exact.

∞ T.P.T F s.t. , .

→ Assume a func. F s.t. .

i.e.,

So,

Since a func. of , is indep. of .

Therefore,

→ The soln. of exact DE is .

→ Reduce a non-exact DE into exact by mult. Int. factor (IF).

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Theorem

Consider the DE,

where , are part. derivs of all points in a rectangular domain .

As if the DE is exact in , then

& conversely, if

then the DE is exact.

Proof

Suppose the DE is exact.

s.t.

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Problems

1. (Verify if exact)

2.

3.