diff eq. transform the given equation into an equivalent system of first order differential equations

y'' + (y')^2 + 25y = sin(x) diff eq. transform the given equation into an equivalent system of first order differential equations

y'' + (y')^2 + 25y = sin(x) @Mathematics

Question

diff eq.  transform the given equation into an equivalent system of first order differential equations

y'' + (y')^2 + 25y = sin(x) diff eq.  transform the given equation into an equivalent system of first order differential equations

y'' + (y')^2 + 25y = sin(x) @Mathematics

anonymous · Answer

anything?

anonymous · Answer

i got nothing

anonymous · Answer

feel free to throw up an airball

amistre64 · Answer

nothing i can be sure about, im just learning to solve these

amistre64 · Answer

y'' = -y'(x)^2 -25y(x) + sin(x) is all i got

amistre64 · Answer

$$\int y'' + (y')^2 + 25y\ dx= \int sin(x)dx$$
$$y' + \frac{1}{3y'}(y')^3 + 25xy= -cos(x)$$
that doesnt look like anything useful

anonymous · Answer

kinda looks like the question again lol

anonymous · Answer

where's JamesJ

amistre64 · Answer

its a mess is what it is.  ive been reading these things for a week and only retain a small portion of it.

james is around, he just answered a q of mine

anonymous · Answer

yea i got a test on wednesday...i need to get crackin at understanding this

jamesj · Answer

Diff eqns are honey for me; I'm surprised I didn't smell this one out 20 minutes ago.

anonymous · Answer

well then great...i got 1 more after this one :)

jamesj · Answer

So, let y1 = y and y2 = y'.

Then 

y1' = y2      ---- (*)

and y2' = y''

hence
y'' + (y')^2 + 25y = sin(x)
is equivalent to
y2' = - y2^2 - 25y1 + sin x   ---- (**)

Therefore the system, combining (*) and (**) is ...

jamesj · Answer

$$\left(\begin{matrix}y_1 \ y_2\end{matrix}\right)' = \left(\begin{matrix}y_2 \ -y_2^2 - 25y_1 + \sin x\end{matrix}\right)$$

jamesj · Answer

It doesn't gain you a lot writing it in column vector form as I just have above.  

But most of the time when you're looking at systems they will be linear, and it is very useful to think of the system this way as a vector ODE, so I'm writing it that way here with one eye to that fact.

anonymous · Answer

can i write the answer like u have written?

jamesj · Answer

Sure.  In fact, I recommend it, so you get use to this sort of notation

anonymous · Answer

wanna do another?

anonymous · Answer

i'll creat another post so i can medal u up again

jamesj · Answer

Yes ... post it on the left that's ok.