Question

Discuss the difference between the differential equations (dy/dx)² = 4y and dy/dx = 2√y. Do they have the same solution curves? Why or why not? Determine the points (a, b) in the plane for which the initial value problem y ′ = 2√y, y(a) = b has (a) no solution, (b) a unique solution, (c) infinitely many solutions.

Answer 1

The solution curves for dy/dx = 2√ y will be solution curves for (dy/dx)² = 4y. However, so will the solution curves for dy/dx = −2√y. So, no, they do not have the same solution curves. The function 2√y is continuous for y ≥ 0. Its partial derivative with respect to y, 1/√y , is continuous for y > 0.

The differential equation dy/dx = 2√y is separable, and we can solve it:

Solving this for y we get

y = (x + C)²  

(a) - There will be no solution if b < 0. 

(b) - There will be (locally) a unique solution for b > 0.

 (c) - If b = 0 there will be infinitely many solutions.