Final answer:
To sketch orthogonal trajectories x²y²=2cy, you can solve the related differential equation. Take the derivative of the given equation with respect to x, rearrange the equation to solve for y', and then solve the separable differential equation to find the equation for the orthogonal trajectories.
Step-by-step explanation:
To sketch orthogonal trajectories x²y²=2cy, we can use the method of solving a differential equation. Start by taking the derivative of the given equation with respect to x, which yields 2xy² + 2x²yy' - 2cy' = 0. Rearrange the equation to solve for y', giving y' = (cy - xy²)/(x²y - c). Now, we can solve this separable differential equation to find the equation for the orthogonal trajectories. Once you have the equation, you can plot the orthogonal trajectories on a graph.