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.

Differentiating ln(x2 + 11) + csc(y) = C we get 2x/ x2 + 11 (__) dy/dy=0 or
2x sin2(y) dx + (______)dy = 0.
Find the constant solutions, if any, that were lost in the solution of the differential equation. (Let k represent an arbitrary integer.)
y =_______

Answer 1

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  = (2x)/( x^2 + 11)  + [-cot * csc(y) ](dy)/(dx)`

=>
`0 = (2x)/( x^2 + 11) + [-(cos(y))/( sin(y) )* (1)/(sin(y)) ](dy)/(dx)`

=>
`0 = (2x)/( x^2 + 11) + [-(cos(y))/( sin^2(y) )](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.`

Explanation: