Differential Equations SOS PLEASE

- anonymous

Differential Equations SOS PLEASE

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- anonymous

\[y'''+y'=\frac{ sinx }{ \cos^2x }\]

- anonymous

\[K^3+K=0\]

- anonymous

\[K(K+1)=0\]

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## More answers

- anonymous

meaning \[K1=0, K2,3= +-i\]

- anonymous

\[Y=C1+C2\cos(x)+C3\sin(x)+Yp\]

- anonymous

now how do I find Yp?

- anonymous

should I find the derivative ?

- ganeshie8

familiar with variation of parameters ?

- anonymous

yes, more or less

- anonymous

@ganeshie8 ?

- anonymous

I really need help with this one

- ganeshie8

\[y'''+y'=\frac{ \sin x }{ \cos^2x }\]
Let \( y'=v\), then the DE becomes
\[v''+v=\frac{ \sin x }{ \cos^2x }\]
which is a familiar second order ordinary eqn, you can find \(Y_p\) using wrokskian or any other tricks that you're familiar with

- anonymous

hmmmm not sure I'm following

- anonymous

I found the general solution

- anonymous

All I need now is the private one

- anonymous

Are you familiar with the method I'm using?

- ganeshie8

http://gyazo.com/85b4d43f26ed36b347d32629ee0df5b0

- ganeshie8

\[v''+v=\frac{ \sin x }{ \cos^2x }\]
earlier you worked the general solution,
general solution for above reduced DE is \(c_2\cos x+c_3\sin x\), yes ?

- anonymous

+C1

- ganeshie8

No, we're only looking at reduced DE for now.

- anonymous

oh ok

- anonymous

Can you please show how to do it using the formula you've attached?

- ganeshie8

Okay, so from general solution we have
\(y_1 = \cos x\)
\(y_2 = \sin x\)
Wronkskian\(W(y_1, y_2) = \begin{vmatrix} \cos x &\sin x\\\cos'x&\sin'x\end{vmatrix} = \cos^2x+\sin^2x = 1 \)

- ganeshie8

simply plug it in the formula and evaluate the integral(s)

- anonymous

but what is g(t)?

- anonymous

whatit refers to?

- ganeshie8

g(t) is whatever there on the right hand side

- ganeshie8

\(g(\color{red}{x}) = \dfrac{\sin x}{\cos^2x}\)

- anonymous

and how do I find C1 then?

- ganeshie8

we will worry about that in the very end

- anonymous

oh ok

- ganeshie8

Keep in mind we're solving the "reduced" DE in \(v\)'s completely first, then we're gona substitute back \(v\)

- anonymous

roger that

- anonymous

I think I can handle with that by myself, but how then can I find C1?

- anonymous

btw thanks a lot!

- ganeshie8

\(\large v(x) = c_2\cos x + c_3\sin x+Y_p\)
plug in \(v(x) = y'\) back

- ganeshie8

and integrate

- anonymous

thank you so much @ganeshie8 !

- anonymous

I think I've got it! I'll try at home and see if it works out for me

- ikram002p

good question, it was useful for me as well .

- ganeshie8

you may refer to this solution generated by wolfram if you get stuck..

##### 1 Attachment

- UsukiDoll

Y_h portion is always easier than finding the Y_p

- UsukiDoll

there are 3 different ways to solve 2nd order odes... method of undetermined coefficients, variation of parameters, and laplace transform. It's just that sometimes one method is easier to use than the other 2. >_<

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