Posted: January 10th, 2017

Prove by using the existence-uniqueness theorem that the following linear initial-value problem dy dt = −p(t)y + q(t), y(t0) = y0, t0 ∈ (a, b) has a unique solution in the interval (a, b) provided that both of the functions p(t), q(t) are continuous in (a, b)

Prove by using the existence-uniqueness theorem that the following linear initial-value problem dy dt = −p(t)y + q(t), y(t0) = y0, t0 ∈ (a, b) has a unique solution in the interval (a, b) provided that both of the functions p(t), q(t) are continuous in (a, b).

[3] (ii) Check if the conditions of the existence-uniqueness theorem are satisfied for following initialvalue problem dy dt = y 2/3 , y(0) = 0. Then solve the above initial-value problem and compare it with what you obtained from the implementation of the existence-uniqueness theorem. [3] (iii) Consider the differential equation dy dt = αy − y 3 , where α is a real parameter. Find the equilibrium solutions, draw the phase line and determine their stability for each value of the parameter. Then draw the bifurcation diagram for the above differential equation.