How to find a first order differential equation for the family of curves?

Question

y=x^(1/2)+cx

 

I can't seem to get the right answer :(

Answer 1

Try differentiating both sides, and you will get a differential equation.

 

dy/dx = (1/2)x^-1/2 + c

 

Assume the boundary condition 

 

y(0) = 0.

 

Then its general solution will be

 

y = x^1/2 + cx + c₁

 

y(0) = 0 ⇒ c₁ = 0

 

Final solution: 

 

y = x^1/2 + cx