The solution of the differential equation $(k x - y^{2}) d y = (x^{2} - k y) d x$ is

x^{3} - y^{3} = 3 k x y + C

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x^{3} + y^{3} = 3 k x y + C

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x^{2} - y^{2} = 2 k x y + C

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x^{2} + y^{2} = 2 k x y + C

No worries! We‘ve got your back. Try BYJU‘S free classes today!

x^{3} - y^{2} = 3 k x y + C

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Solution

The correct option is A $x^{3} + y^{3} = 3 k x y + C$
We have $(k x - y^{2}) d y = (x^{2} - k y) d y$
$\Rightarrow k (x d y + y d x) = x^{2} d y + y^{2} d x$
$\Rightarrow k d (x y) = \frac{1}{3} d (x^{3} + y^{3})$
Integrating on both sides, we get
$k x y = \frac{x^{3} + y^{3}}{3} + C_{1}$
$\Rightarrow x^{3} + y^{3} = 3 k x y + C$

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The solution of the differential equation (kx−y2)dy=(x2−ky)dx is

The correct option is A x3+y3=3kxy+CWe have (kx−y2)dy=(x2−ky)dy⇒k(xdy+ydx)=x2dy+y2dx⇒kd(xy)=13d(x3+y3)Integrating on both sides, we getkxy=x3+y33+C1⇒x3+y3=3kxy+C

The solution of the differential equation $(k x - y^{2}) d y = (x^{2} - k y) d x$ is