Solution:

Given differential equation is y * y' = sec(x)

To find the solution y(x), we need to solve this differential equation. We can do this by separating the variables and integrating both sides.

Separating the variables, we get:

y * dy = sec(x) * dx

Integrating both sides, we have:

∫ y * dy = ∫ sec(x) * dx

Let's solve each side of the equation separately.

∫ y * dy:

To integrate y * dy, we can use the substitution method. Let's substitute y = u(x) * sin(sin(x)) * v(x) * cos(x), where u(x) and v(x) are functions of x.

Differentiating y with respect to x, we get:

y' = u' * sin(sin(x)) * v * cos(x) + u * cos(sin(x)) * sin(x) * v' * cos(x) - u * sin(sin(x)) * v * sin(x) * sin(x) - u * sin(sin(x)) * v * sin(x) * cos(x)

Simplifying the above expression, we have:

y' = u' * sin(sin(x)) * v * cos(x) + u * cos(sin(x)) * sin(x) * v' * cos(x) - u * sin^2(sin(x)) * v * sin(x) - u * sin^2(sin(x)) * v * cos(x)

Now, substituting these values of y and y' in the differential equation, we get:

(u * sin(sin(x)) * v * cos(x)) * (u' * sin(sin(x)) * v * cos(x) + u * cos(sin(x)) * sin(x) * v' * cos(x) - u * sin^2(sin(x)) * v * sin(x) - u * sin^2(sin(x)) * v * cos(x)) = sec(x)

Expanding the above equation and simplifying, we get:

u^2 * sin^2(sin(x)) * v^2 * cos^2(x) * (u' * sin(sin(x)) * v * cos(x) + u * cos(sin(x)) * sin(x) * v' * cos(x) - u * sin^2(sin(x)) * v * sin(x) - u * sin^2(sin(x)) * v * cos(x)) = sec(x)

Now, let's solve the other side of the equation.

∫ sec(x) * dx:

The integral of sec(x) can be found using the logarithmic identity:

∫ sec(x) * dx = ln|sec(x) + tan(x)| + C1

where C1 is the constant of integration.

Now, let's combine both sides of the equation and solve for u(x) and v(x).

u^2 * sin^2(sin(x)) * v^2 * cos^2(x) * (u' * sin(sin(x)) * v * cos(x) + u * cos(sin(x)) * sin(x) * v' * cos(x) - u * sin^2(sin(x)) * v * sin(x) - u * sin^2(sin(x)) * v * cos(x)) = ln|sec(x) + tan(x)| + C1

Since the left-hand side of the equation is a function of x, and the right-hand side is a constant