The discussion focuses on solving the differential equation dy/dx = 4x - 2y by finding constants a and b in the linear equation y = ax + b. The solution process reveals that for the equation to hold for all x, the condition 4 - 2a = 0 must be satisfied, leading to a = 2 and b = -1. The relationship a = -2b is established, indicating that once a is chosen, b can be directly determined. This analysis provides a clear method for solving similar linear differential equations.
PREREQUISITESStudents studying calculus, mathematics educators, and anyone interested in mastering differential equations and their applications.
##NihalRi said:a = (4-2a) x - 2b (what happened here?)
BvU said:##
a=4x-2 (ax+b) \Leftrightarrow a=4x-2ax-2 b \Leftrightarrow a= (4-2a) x-2 b ## can only be true for all x if ##(4-2a)=0##. What remains is ## a = -2b ## . Pick an a and a b follows.