Question

Solve the differential equation:

Answer 1

\[\frac{ d^2y }{ dx^2 }+2\frac{ dy }{ dx }+2y=10\]

Answer 2

Laplace transforms

Answer 3

\[\large y'' +y'+2y = 10\] how do you do laplace transformations..... :|

Answer 4

Wait NVM LOL Do you have any initial conditions?

Answer 5

YEAH got you there. :|

Answer 6

what's that?

Answer 7

what's what?

Answer 8

Laplace transformations..

Answer 9

Idk....

Answer 10

or laplace transforms..

Answer 11

\(y''++y'+2y=10\)
1) Homogeneous equation: \(y''+y+2y=0\) (H)
-> \(p(\lambda) = \lambda^2 + \lambda + 2=0\)
iff \(\lambda=\frac{-1\pm i\sqrt7}{2} = -\frac12 \pm i\frac{\sqrt7}2\) (i think)
The functions \(g(x) = e^{-\frac12x}(A...+B...)\), with \(A,B\in\mathbb R\) are solutions of (H).
2) find a particular solution.
It seems that one constant function is solution of this. (\(y(x) = c\) for some \(c\)).
3) write the general form of the solution.

This method is quite common.

Answer 12

@reemii that looks way too complicated...

Answer 13

it's a method, no need to think. It is likely the asker learned the method. ( @Luigi0210 )

Answer 14

hm, from what math is it?

Answer 15

what do you mean?

Answer 16

Is this method from calculus 2?

Answer 17

no idea.

Answer 18

@Loser66 from \(\lambda=-\frac12\pm i \sqrt{7}/2\) you write \(u=-\frac12,v=\sqrt{7}/2\).

then (complete version) \(g(x)=e^{ux}(A\cos(vx)+B\sin(vx))\).

A solution of the initial equation is \(y(x)=5\).

General solution: \(y_\text{gen}(x)=5+g(x)\), \(A,B\in\mathbb R\).

Answer 19

I'll look into the laplace transorm method. :)

Answer 20

Can't use laplace transforms here without initial conditions. Sry :(

Answer 21

@Loser66 Convert your 2nd order ODE into two 1st order ODE's

Answer 22

@FutureMathProfessor the same with my method.need initial conditions

Answer 23

@FutureMathProfessor You know that I didn't take differential equation yet, how can I know what is Laplace transform? I use my own knowledge only.

Answer 24

@Loser66 This is my DiffEQ test from a VERY similar problem. No initial conditions needed
http://gyazo.com/e8615a0188dd35317e433142262c032c.png