Answer to Question #288118 in Differential Equations for Eybc

Question #288118

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

1
Expert's answer
2022-01-17T18:07:12-0500

Solution:

4(D+3)y=0(D+3)y=0(ddx+3)y=0dydx+3y=0dydx=3ydyy=3dx4(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,

logy=3x+Cy=e3x+Cy=C1e3x, where C1=eC\log y=-3x+C \\ \Rightarrow y=e^{-3x+C} \\ \Rightarrow y=C_1e^{-3x}, \ where \ C_1=e^C


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