Question

Show that an implicit solution of 2x sin2(y) dx - (x2 + 11) cos(y) dy = 0 is given by ln(x2 + 11) + csc(y) = C.

Answer 1

Answer:

From the question we are told that

  The  equation is  $2x sin^2(y) dx - (x^2 + 11) cos(y) dy = 0$

Generally the goal of this solution is to prove that the implicit solution of the differential  equation above is  $ln(x^2 + 11) + csc(y) = C.$

First Step  

    Differentiate the solution with respect to  x  

   $0 = \frac{2x}{ x^2 + 11} + [-cot * csc(y) ]\frac{dy}{dx}$

=> $0 = \frac{2x}{ x^2 + 11} + [-\frac{cos(y)}{ sin(y) }* \frac{1}{sin(y)} ]\frac{dy}{dx}$  

=>  $0 = \frac{2x}{ x^2 + 11} + [-\frac{cos(y)}{ sin^2(y) }]\frac{dy}{dx}$

=>  

multiply through by  $sin^2(y)$ , $x^2 + 11$, $dx$

So

=> $2 x (sin^2(y) dx + [-cos(y) (x^2 + 11)]dy = 0$

Looking at the equation obtained we see that it is equivalent to the differential  equation hence  the implicit solution of $2x sin^2(y) dx - (x^2 + 11) cos(y) dy = 0$ is   $ln(x^2 + 11) + csc(y) = C.$

Step-by-step explanation: