1. ## Differential Equation

Hi everybody,

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

2. Originally Posted by woody198403
Hi everybody,

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
Read this: PlanetMath: second order linear differential equation with constant coefficients

3. Actually, I think I know how to do this. After I finish what I think is the answer, and if anybody can let me know if Im right or wrong, that would be great. Thanks

4. Originally Posted by woody198403
Actually, I think I know how to do this. After I finish what I think is the answer, and if anybody can let me know if Im right or wrong, that would be great. Thanks
You can check your answer yourself simply by substituting it into the differential equation.

5. 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?

6. 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?

7. Originally Posted by woody198403
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 $\displaystyle \left( m + \frac{3}{2} \right)^2 = -\frac{11}{4} \Rightarrow m + \frac{3}{2} = \pm \sqrt{\frac{-11}{4}} = \pm \frac{\sqrt{-11}}{2} = \pm \frac{i \, \sqrt{11}}{2}$ .......

8. I see where I slipped up. thanks