Final answer:
The differential equation is solved by separating variables and integrating both sides. The solution is expressed as y = √(12x + 2C), where C is the constant of integration and represents a family of solutions.
Step-by-step explanation:
Solving the Differential Equation
The provided differential equation is dy/dx ycos(x) = 6cos(x). This equation can be simplified by separating the variables, allowing us to integrate both sides. Recognizing that cos(x) appears on both sides of the equation, we can simplify to dy/dx y = 6. Integrating both sides with respect to x, we get:
- Integral of y dy = Integral of 6 dx
- (1/2)y2 = 6x + C
- y2 = 12x + 2C (where C is the constant of integration)
- y = √(12x + 2C)
We then solve for the constant C using the initial condition provided in the context of the question (if given) to find the particular solution. As no initial conditions were specified, the solution will contain the arbitrary constant C, representing the family of solutions to the differential equation.