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Solution of the differential equation $y - x \frac{d y}{d x} = a (y^{2} + \frac{d y}{d x})$

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a

x+a1−ay=cy

b

x−a1−ay=cy

c

x−a1−ay=cx

d

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detailed solution

Correct option is A

We have y−xdydx=ay2+dydx⇒ydx−xdy=ay2dx+ady⇒y(1−ay)dx=(x+a)dy⇒dxx+a−dyy(1−ay)=0 since 1y1-ay=1y+a1-ay Integrating , we getlog⁡(x+a)-log⁡y-alog⁡(1−ay)-a=log⁡clog (x+a)-log y+log(1-ay)=logc⇒log⁡(a+x)(1−ay)y=log⁡c⇒(x+a)(1−ay)=cy

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The general solution of the differential equation $(1 + y^{2}) d x + (1 + x^{2}) d y = 0$ is

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