Question

PARTA
5.
Under what condition the equation (AX + By) dx + (Cx+Dy) dy=0 is exact​

Answer 1

SOLUTION

TO DETERMINE

The condition for which the equation (AX + By) dx + (Cx+Dy) dy=0 is exact

CONCEPT TO BE IMPLEMENTED

An equation of the form Mdx + Ndy = 0 is exact if

$\displaystyle \sf{ \frac{ \partial M}{ \partial y} = \frac{ \partial N}{ \partial x} }$

EVALUATION

Here the given equation is

(AX + By) dx + (Cx+Dy) dy=0

Comparing with Mdx + Ndy = 0 we get

M= (AX + By) & N = (Cx+Dy)

So the equation is exact if

$\displaystyle \sf{ \frac{ \partial M}{ \partial y} = \frac{ \partial N}{ \partial x} }$

$\displaystyle \sf{ \implies \: \frac{ \partial }{ \partial y}(Ax + By) = \frac{ \partial}{ \partial x}(Cx + Dy) }$

$\displaystyle \sf{ \implies \: 0 + B = C +0}$

$\displaystyle \sf{ \implies \: B = C}$

FINAL ANSWER

The required condition is B = C

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