Question

The solution of dydx=xlog⁡x2+xsin⁡y+ycos⁡y

• A. (A) y sin y = x2 log x + c
• B. (B) y sin y = x2 + c
• C. (C) y sin y = x2 + log x + c
• D. (D) y sin y = x log x + c

Answer 1

Answer: (A)

(A) y sin y = x2 log x + c

Answer 2

Explanation:
Given equation is dydx=xlog⁡x2+xsin⁡y+ycos⁡y⇒(sin⁡y+ycos⁡y)dy=(xlog⁡x2+x)dxOn integrating both sides, we get ∫(sin⁡y+ycos⁡y)dy=∫(x⋅log⁡x2+x)dx⇒−cos⁡y+ysin⁡y+cos⁡y=x22log⁡x2−∫x22⋅1x22xdx+∫xdx+c⇒ysin⁡y=x22⋅2log⁡x−∫xdx+∫xdx+c⇒ysin⁡y=x2log⁡x+c