Answer to Question #181756 in Differential Equations for Muskan

Question

d²y/d²x+2dy/dx+10y=0

Answer 1

$\frac{d^{2}y}{dx}+2\frac{dy}{dx}+10y=0$

Let $y=e^{mx}$

Then $\frac{dy}{dx}=me^{mx}$ and $\frac{d^{2}y}{dx^{2}}=m^{2}e^{mx}$

Therefore we get $(m^{2}+2m+10)e^{mx}=0$

As $e^{mx}\neq 0,$ we have $(m^{2}+2m+10)=0$

i.e., $m=\frac{-2\pm \sqrt{ 4-40}}{2}$

$=\frac{-2\pm i6}{2}$

$=-1\pm i3$

Therefore the solution becomes $y=e^{-1}(c_{1} Cos 3x+c_{2} Sin 3x)$ where $c_{1},c_{2}$ are constants.