If the function y=e4x+2e−x is a solution of the differential equation d3ydx3−13dydxy=k then K is

y=e4x+2e−x ⇒y2-3y1-4=0 ∵y2-(α+β)y1-αβy=0 ⇒y3-3y2-4y1=0 ⇒y3-3(3y1+4y)-4y1=0 ⇒y3-9y1-12y-4y1=0 ⇒y3−13y1=12y ⇒y3−13y1y=12

If the function y=e4x+2e−x is a solution of the differential equation d3ydx3−13dydxy=k then K is

y=e4x+2e−x ⇒y2-3y1-4=0 ∵y2-(α+β)y1-αβy=0 ⇒y3-3y2-4y1=0 ⇒y3-3(3y1+4y)-4y1=0 ⇒y3-9y1-12y-4y1=0 ⇒y3−13y1=12y ⇒y3−13y1y=12

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If the function $y = e^{4 x} + 2 e^{- x}$ is a solution of the differential equation $\frac{\frac{d^{3} y}{d x^{3}} - 13 \frac{d y}{d x}}{y} = k$ then K is

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Detailed Solution

$y = e^{4 x} + 2 e^{- x} \Rightarrow y_{2} - 3 y_{1} - 4 = 0 (∵ y_{2} - (α + β) y_{1} - α β y = 0) \Rightarrow y_{3} - 3 y_{2} - 4 y_{1} = 0 \Rightarrow y_{3} - 3 (3 y_{1} + 4 y) - 4 y_{1} = 0 \Rightarrow y_{3} - 9 y_{1} - 12 y - 4 y_{1} = 0 \Rightarrow y_{3} - 13 y_{1} = 12 y \Rightarrow \frac{y_{3} - 13 y_{1}}{y} = 12$