Ordinary Differential Equation

Question 4. Let t,:c E R and 1603,53), aftax) a," Assume f (t, :3) satises | f (t, w)| g A+B|x| for any a: E R, and A, B are positive constants. be continuous functions on R2. d Prove that the ODE $ = f(t, a") has a solution a: = @(t) globally in time for dt t e (-00, +00). (Hint: try to choose appropriate :3, b in the local existence and uniqueness theory, so that the step size h for the local existence has a uniform positive lower bound ho depending only on A and B.)