Question

The differential equation (x+y+2)dy/dx+x(x+y-1)=0 is
(A) variable seprable
(B) linear
(C) homogeneous
(D) exact​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

The differential equation

$\displaystyle \sf{(x + y + 2) \frac{dy}{dx} + (x + y - 1) = 0}$

(A) variable separable

(B) linear

(C) homogeneous

(D) exact

EVALUATION

Here the given differential equation is

$\displaystyle \sf{(x + y + 2) \frac{dy}{dx} + (x + y - 1) = 0}$

Which can be rewritten as

$\displaystyle \sf{(x + y + 2) dy + (x + y - 1) dx= 0}$

$\displaystyle \sf{ \implies \: (x + y - 1) dx + (x + y + 2) dy = 0}$

Which is of the form Mdx + Ndy = 0

Where

M = x + y - 1

N = x + y + 2

Now

$\displaystyle \sf{ \frac{ \partial M }{ \partial y} = 1}$

$\displaystyle \sf{ \frac{ \partial N }{ \partial x} = 1}$

Therefore

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

So the given differential equation is exact

FINAL ANSWER

Hence the correct option is (D) exact

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