Hi everybody,
Can anybody please help me solve the following differential equation:
d^2y/dx^2 + 3(dy/dx) +5y = 0, y(0) = 2, dy/dx(0) = 0
Even a a simple guide to solving this would be great.
Cheers
Hello, woody198403!
$\displaystyle \frac{d^2y}{dx^2} + 3\frac{dy}{dx} +5y \:=\: 0,\quad y(0) \,= \,2,\quad\frac{dy}{dx}(0) \,= \,0$
The characteristic equation is: .$\displaystyle m^2 + 3m + 5 \:=\:0$
. . Its roots are: .$\displaystyle m \:=\:\frac{-3 \pm 4i}{2} \:=\:-\frac{3}{2} \pm 2i$
The general solution is: .$\displaystyle y \;=\;e^{-\frac{3}{2}x}\bigg(C_1\cos 2x + C_2\sin 2x\bigg)$
Can you finish it?
How did you get the roots? Im a bit rusty with complex numbers, but you've lost me.
I cannot get the same answer as you.
here is what ive done:
m^2 + 3m + 5 = 0
so
(m+3/2)^2 = -11/4
so
m = -3/2 + or - sqrt(-11/4)
m = -3/2 + or - (11i/4)
How did you get m= -3/2 + or - 2i?
Well, Soroban slipped up (as can happen) and you've slipped up too. Using the quadratic formula:
$\displaystyle m = \frac{-3 \pm \sqrt{-11}}{2} = - \frac{3}{2} \pm i \, \frac{\sqrt{11}}{2}$.
Note that $\displaystyle \left( m + \frac{3}{2} \right)^2 = -\frac{11}{4} \Rightarrow m + \frac{3}{2} = \pm \sqrt{\frac{-11}{4}} = \pm \frac{\sqrt{-11}}{2} = \pm \frac{i \, \sqrt{11}}{2}$ .......