Answer:
From the question we are told that
The equation is [tex]2x sin^2(y) dx - (x^2 + 11) cos(y) dy = 0[/tex]
Generally the goal of this solution is to prove that the implicit solution of the differential equation above is [tex]ln(x^2 + 11) + csc(y) = C.[/tex]
First Step
Differentiate the solution with respect to x
[tex]0 = \frac{2x}{ x^2 + 11} + [-cot * csc(y) ]\frac{dy}{dx}[/tex]
=> [tex]0 = \frac{2x}{ x^2 + 11} + [-\frac{cos(y)}{ sin(y) }* \frac{1}{sin(y)} ]\frac{dy}{dx}[/tex]
=> [tex]0 = \frac{2x}{ x^2 + 11} + [-\frac{cos(y)}{ sin^2(y) }]\frac{dy}{dx}[/tex]
=>
multiply through by [tex]sin^2(y)[/tex] , [tex] x^2 + 11[/tex], [tex]dx[/tex]
So
=> [tex]2 x (sin^2(y) dx + [-cos(y) (x^2 + 11)]dy = 0[/tex]
Looking at the equation obtained we see that it is equivalent to the differential equation hence the implicit solution of [tex]2x sin^2(y) dx - (x^2 + 11) cos(y) dy = 0[/tex] is [tex]ln(x^2 + 11) + csc(y) = C.[/tex]
Step-by-step explanation:
