What is the general solution of the differential equation x d y - y dx = y2 ?Where c is an arbitrary constanta)x = cyb)y2 = cxc)x +xy - cy = 0d)None of theseCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

SOL:- Given xdy-ydx=y^2
( xdy-ydx)/y^2=0
(ydx-xdy)/y^2=0
d(x/y)=0
x/y=integration(0)
x/y=0+c
x=cy

Answer

To find the general solution of the given differential equation, we can rearrange the equation as follows:

xdy - ydx = y^2

Dividing both sides by y^2, we get:

(x/y^2)dy - dx/y = 0

Now, let's rewrite the equation in a standard form:

(x/y^2)dy = dx/y

Separating the variables, we can write:

y^2dy = xdx

Integrating both sides, we get:

∫ y^2dy = ∫ xdx

This gives us:

(1/3)y^3 = (1/2)x^2 + c

where c is the constant of integration.

Now, let's solve for y in terms of x:

y^3 = (3/2)x^2 + 3c

Taking the cube root of both sides, we get:

y = (3/2)^(1/3) * x^(2/3) + c^(1/3)

Thus, the general solution of the given differential equation is:

y = (3/2)^(1/3) * x^(2/3) + c^(1/3)

where c is an arbitrary constant.

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