Question

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

• A. 0
• B. 1
• C. 2
• D. more than 2

Answer 1

Answer: (C)

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