Question

Exact Solution of 1st-order system of Differential Equations

Find the Particular solution of the following differential equation with the initial conditions:

ー-3-2y, x(t = 0) = 3; 5x - 4y

pls don't solve this using matrices.

Answer 1

Given a 1st order system of differential equation: - partial differential x/ partial differential t = 3x - 2y: x(t = 0) = 3 & partial differential y/ partial differential t = 5x - 4y: y(t = 0) = 6} (1) we shall solve the given system of differential equations simultaneously we can write (1) in the form: - Dx = 3x - 2y (D - 3)x + 2y = 0 Dy = 5x - 4y (D + 4)y - 5x = 0 where Dx = partial differential x/ partial differential x & Dy = partial differential y/ partial differential t: D being the differential operator So we have: (D - 3)x = - 2y (2) (D + 4)y = 5x (3) using the operator (D + 4) on (2) we get: (D - 3) (D + 4)x = - 2(D + 4)y substituting the value of (D + 4)y = 5x is the above equation we get: (D - 3) (D + 4)x = - 2(5x) (D^2 + 4D - 3D - 12)x = - 10x
As per your requirements, I haven't solved the system of differential equations by matrix method instead I have used the operator method to solve the simultaneous system of differential equations. If you find my answer helpful please do give me a thumbs up vote on my answer. For any other doubts or queries relating to the solution, please feel free to ask. Thanks!