How To Find The Orthogonal Trajectories | x^2 +y^2 = c^2

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Understanding a family of curves and their trajectories
Given a family of curves, like y=kxy=kx, we can choose different values for kk to write the equations of some of the curves in the family.

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Population Dynamics
OR
exponential model

The population of a certain community is known to increase at a rate proportional
to the number of people present at any time. The population has doubled in 5 years,
how long would it take to triple?. If it is known that the population of the
community is 10, 000 after 3 years. What was the initial population? What will be
the population in 30 years?

Student.