Mathematics, 11.02.2020 19:50 bob4059
Show that an implicit solution of 2x sin2(y) dx − (x2 + 12) cos(y) dy = 0 is given by ln(x2 + 12) + csc(y) = C. Differentiating ln(x2 + 12) + csc(y) = C we get 2x x2 + 12 + dy dx = 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 =
Answers: 3
and 
:![2x\sin (2y)dx=(x^2+12)\cos y dy\\ \\\dfrac{2x\sin (2y)dx}{x^2+12}=\cos ydy\ [\text{Divided by non-zero expression }x^2+12]\\ \\\dfrac{2x}{x^2+12}dx=\dfrac{\cos y}{\sin (2y)}dy\ [\text{Divided by }\sin (2y)]\\ \\\dfrac{2x}{x^2+12}dx=\dfrac{\cos y}{2\sin y\cos y}dy\ [\text{Use formula }\sin (2y)=2\sin y\cos y]\\ \\\dfrac{2x}{x^2+12}dx=\dfrac{1}{2\sin y}dy\ [\text{Simplify when }\cos y\neq 0]](/tpl/images/0506/8830/fdc40.png)
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Show that an implicit solution of 2x sin2(y) dx − (x2 + 12) cos(y) dy = 0 is given by ln(x2 + 12) +...
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