(tan(x) − Sin(x) Sin(y)) Dx + Cos(x) Cos(y) Dy = 0

Jasonbradley

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Jun 16, 2024

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Solving the Differential Equation: (tan(x) − sin(x) sin(y)) dx + cos(x) cos(y) dy = 0

This differential equation is of the form M(x, y) dx + N(x, y) dy = 0, where:

• M(x, y) = tan(x) − sin(x) sin(y)
• N(x, y) = cos(x) cos(y)

To solve this, we can check if it's an exact differential equation. This means that the partial derivative of M with respect to y should be equal to the partial derivative of N with respect to x. Let's check:

• ∂M/∂y = -sin(x) cos(y)
• ∂N/∂x = -sin(x) cos(y)

Since ∂M/∂y = ∂N/∂x, the equation is exact. This means there exists a function F(x, y) such that:

• ∂F/∂x = M(x, y)
• ∂F/∂y = N(x, y)

To find F(x, y), we integrate M(x, y) with respect to x, treating y as a constant:

F(x, y) = ∫ (tan(x) − sin(x) sin(y)) dx = ln|sec(x)| + cos(x) sin(y) + g(y)

Here, g(y) is an arbitrary function of y. Now, we differentiate F(x, y) with respect to y and compare it to N(x, y):

∂F/∂y = cos(x) cos(y) + g'(y)

Comparing this with N(x, y) = cos(x) cos(y), we find that g'(y) = 0. Integrating this gives us:

g(y) = C, where C is an arbitrary constant.

Therefore, the general solution to the differential equation is:

F(x, y) = ln|sec(x)| + cos(x) sin(y) + C = 0

This solution represents a family of curves that are implicitly defined by the equation.