Let y₀ > 0, z₀ > 0 and α > 1.

Consider the following two differential equations:

We say that the solution to a differential equation exists globally if it exists for all t > 0.

Which of the following statements is true?

Explanation:

and y(0) = y₀ ⇒ 

then y1-α = (1 - α)t + y01-α

and If α > 1 ⇒ 1 - α < 0. suppose 1 - α = -a, a > 0

then (1) y-a = -at + y0-a 

⇒ 

then for y0-a - at = 0, solution does not exist.

(∵ y0 > 0, a > 0) 

∴ (*) has no global solution

as  ∈ (0, ∞)

options (1) and (3) are false

(* *)  ⇒ 

and z0 = z(0) ⇒ 

∴ z1-α = -(1 - α)t + z01-α ....(ii)

and for α > 1 ⇒ 1 - α < 0. So, Suppose 1 - α = - b, b > 0

then (ii) ⇒ z-b = bt + zo-b ⇒ 

and for bt + z0-b​ = 0 solution does not exist 

⇒  < o

(∵ zo > o, b > 0)

So, ∀ t > 0, solution exist of (* *)

⇒​ (**) have global solutions.

option (2) is false.

Also, for T < ∞, take T = 

= + ∞ 

∴ option (4) is true.