Question

Solve the differential equation \( (x - \sin y) \, dy + (\tan y) \, dx = 0 \), given \( y(0) = 0 \).

Hint:

For solving separable differential equations, ensure that the integrals are correctly computed and that initial conditions are applied to find the constant of integration.

Answer 1

We are given the differential equation: \[ (x - \sin y) \, dy + (\tan y) \, dx = 0 \] Rearrange the terms to separate the variables \( x \) and \( y \): \[ (x - \sin y) \, dy = - (\tan y) \, dx \] Now, separate the variables \( x \) and \( y \): \[ \frac{dy}{\tan y} = - \frac{dx}{x - \sin y} \] This gives the following integral for each side: For the left-hand side, we use the fact that: \[ \int \frac{1}{\tan y} \, dy = \ln |\sin y| \] And for the right-hand side: \[ \int \frac{1}{x - \sin y} \, dx \] This integral is straightforward and yields: \[ \ln |x - \sin y| \] Thus, the general solution is: \[ \ln |\sin y| = - \ln |x - \sin y| + C \] where \( C \) is the constant of integration. Exponentiate both sides to get rid of the logarithms: \[ |\sin y| = \frac{C}{|x - \sin y|} \] Now, use the initial condition \( y(0) = 0 \) to find \( C \): \[ \sin(0) = \frac{C}{|0 - \sin 0|} \quad \Rightarrow \quad C = 0 \] Thus, the solution to the differential equation is: \[ \boxed{\sin y = 0} \]