Mathematics: 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 =

b = 12Step-by-step explanation:...

Show that an implicit solution of 2x sin2(y)

Ответ:

bunnles

03.02.2022

Mathematics

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 ,

=> 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:

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