Question #7428

Is the following differential equation exact or not and solve it if its exact ?
( x^2 + cosy ) + ( y^2+x+sinx) dy/dx = x

Expert's answer

Differential equation of the form I(x,y)dx + J(x,y)dy = 0 is exact only if dI/dy =dJ/dx .

We can transform our equation to this form :

( x^2 + cosy )dx + ( y^2+x+sinx )dy = xdx

( x^2 - x + cosy )dx + ( y^2+x+sinx )dy = 0 .

In this equation I(x,y) = x^2 - x + cosy and J(x,y) = y^2+x+sinx .

Then we find : dI/dy = -siny and dJ/dx = 1 + cosx.

We see that dI/dy not equal dJ/dx .

That's why given equation is not exact.

We can transform our equation to this form :

( x^2 + cosy )dx + ( y^2+x+sinx )dy = xdx

( x^2 - x + cosy )dx + ( y^2+x+sinx )dy = 0 .

In this equation I(x,y) = x^2 - x + cosy and J(x,y) = y^2+x+sinx .

Then we find : dI/dy = -siny and dJ/dx = 1 + cosx.

We see that dI/dy not equal dJ/dx .

That's why given equation is not exact.

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