Answer to Question #288118 in Differential Equations for Eybc

Question

Obtain the general solutions given the differential operators (D + 3)4 y = 0

Answer 1

Solution:

$4(D + 3) y = 0 \\ \Rightarrow (D + 3) y = 0 \\ \Rightarrow (\dfrac d{dx} + 3) y = 0 \\ \Rightarrow \dfrac {dy}{dx}+3y=0 \\ \Rightarrow \dfrac {dy}{dx}=-3y \\ \Rightarrow \dfrac {dy}{y}=-3dx$

On integrating both sides,

$\log y=-3x+C \\ \Rightarrow y=e^{-3x+C} \\ \Rightarrow y=C_1e^{-3x}, \ where \ C_1=e^C$