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Q.
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 the value of $K$ is :
Differential Equations
Solution:
$y = e ^{4 x }+2 e ^{- x } ; y _1=4 e ^{4 x }-2 e ^{- x } ; y _2=16 e ^{4 x }+2 e ^{- x } ; y _3=64 e ^{4 x }-2 e ^{- x }$
Now, $y _3-13 y _1=\left(64 e ^{4 x }-2 e ^{- x }\right)-13\left(4 e ^{4 x }-2 e ^{- x }\right)=12 e ^{4 x }+24 e ^{- x }$
$=12\left( e ^{4 x }+2 e ^{- x }\right)=12 y$
$\therefore \frac{ y _3-13 y _1}{ y }=12 \Rightarrow D$