Question

The general solution of the equation $\frac{d^{3}y}{d{x}^3}-7\frac{d^{2}y}{d{x}^3}+16\frac{dy}{dx}-12y=0$ is :

• A. $c_1{e}^{2x}+c_2{e}^{-2x}+c_3{e}^{-3x}$
• B. $(c_{1}x+c_2)e^{2x}+c_3{e}^{3x}$
• C. $(c_{1}x+c_2)e^{-3x}+c_3{e}^{2x}$
• D. $(Acosx+Bsinx)e^{2x}+c_3{e}^{3x}$

Answer 1

Answer: (C)

$(c_{1}x+c_2)e^{-3x}+c_3{e}^{2x}$

$\frac{d^{3}y}{d{x}^3}-7\frac{d^{2}y}{d{x}^3}+16\frac{dy}{dx}-12y=0$
Substitute $y=k{e}^{mx}$, since it is a homogeneous equation
$m^3-7m^2+16m-12=0$
$\Rightarrow (m-3)(m-2{)}^2=0$
$\Rightarrow m=3,2,2$
So general solution is
$(c_{1}x+c_2)e^{2x}+c_3{e}^{3x}$