Question

Show that an implicit solution of 2x sin2(y) dx − (x2 + 10) cos(y) dy = 0 is given by ln(x2 + 10) + csc(y) = C. Differentiating ln(x2 + 10) + csc(y) = C we get 2x x2 + 10 + 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.)

Answer 1

Answer:

Step-by-step explanation:

$2xsin(2y)dx-(x^2+10) cosy dy =0\\\\\frac{2x}{x^2 + 10}dx= \frac{cosy}{sin(2y)}$

Take integration both side (apply substitution for the left hand side, apply sin(2y) = 2 sin(y) cos(y) for the right hand side) you will have the condition.

Problem solved