Final answer:
The question involves solving a differential equation with trigonometric functions by substituting v = 1/cosy. Trigonometric relationships from right triangles are used to understand the components of vectors and velocity. The equation is then simplified, and variables are separated for integration to find a solution.
Step-by-step explanation:
The question asks us to solve a differential equation involving trigonometric functions. The given equation is siny(dy/dx) = cosy(1 - xcosy), with a hint to substitute v = 1/cosy. By substituting v into the equation, we can simplify and separate variables to solve the differential equation.
Using the trigonometric identities from a right triangle, Ax/A = cos A relates the adjacent side to the hypotenuse, and Ay/A = sin A relates the opposite side to the hypotenuse. Knowing these relationships, we can use substitutions to find scalar components of vectors. Similarly, the components of velocities have forms such as v cos θ, as shown by the relationships V1x = V1 for a particle moving along the x-axis.
When we apply the given hint to the differential equation, we can rewrite the equation as dy/dx = v(1 - xv), and by further separating variables, we can integrate both sides to find a solution.