Question

The solution of the differential equation dydx=sec⁡(yx)+yx is:

• A. (A) cos⁡(yx)=log⁡(cx)
• B. (B) sin⁡(xy)=log⁡(cx)
• C. (C) sin⁡(yx)=log⁡(cx)
• D. (D) None of these

Answer 1

Answer: (C)

(C) sin⁡(yx)=log⁡(cx)

Answer 2

Explanation:
Given,dydx=sec⁡(yx)+yx……(1) Let,yx=t ⇒y=xtDifferentiating with respect to x, we getdydx=x×dtdx+t×ddxxWe known that:ddxxy=y×ddxx+x×ddxySo,dydx=x×dtdx+tNow, Putting these value in equation (1), we getxdtdx+t=sec⁡t+t⇒xdtdx=sec⁡t⇒dtsec⁡t=dxxIntegrating both sides, we get∫dtsec⁡t=∫dxx⇒∫cos⁡tdt=∫dxx⇒sin⁡t=log⁡x+log⁡c⇒sin⁡t=log⁡(cx)(∵log⁡m+log⁡n=log⁡(mn))∴sin⁡(yx)=log⁡(cx)Hence, the correct option is (C).