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  4. Number of values of m ∈ N for which y =e mx is a solution of the differential equation D3 y-3 D2 y-4 D y+12 y=0 is -

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Q. Number of values of $m \in N$ for which $y =e^{ mx }$ is a solution of the differential equation $D^3 y-3 D^2 y-4 D y+12 y=0$ is -

Differential Equations

Solution:

$y=e^{m x} \text { then } D(y)=m e^{m x}, D^2(y)=m^2 e^{m x}$
$D^3(y)=m^3 e^{m x}$
then given
$D^3 y-3 D^2 y-4 D y+12 y=0$
$\Rightarrow m^3 e^{m x}-3 m^2 e^{m x}-4 m e^{m x}+12 e^{m x}=0 $
$\Rightarrow m^3-3 m^2-4 m+12=0\left(\because e^{m x} \neq 0\right) $
$\Rightarrow(m-2)(m-3)(m+2)=0 $
$\Rightarrow m=2,3,-2$
Hence number of values of $m \in N$ will be 2 .