A small contribution from the rational theory of consumer to solving differential equations first-order inhomogeneous with initial and final conditions

This investigation based on the theory of the rational consumer answers the following question: What condition must function yt
satisfy for the inhomogeneous first-order differential equation t tt x xy  = + α with boundary conditions 0 0y 0 T x x > = to have a unique solution? To do this, a problem of a rational consumer that maximizes his total discounted utility (with his subjective discount rate) on a non-perishable consumer good, in continuous time in a finite planning horizon, subject to the discounted present value (with the real interest rate) of his planned consumption is equal to his initial (positive) wealth and does not leave inheritance.

mathematical methods, partial differential equations, rational consumer