Differential Equations SOS PLEASE

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Differential Equations SOS PLEASE

Mathematics
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\[y'''+y'=\frac{ sinx }{ \cos^2x }\]
\[K^3+K=0\]
\[K(K+1)=0\]

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meaning \[K1=0, K2,3= +-i\]
\[Y=C1+C2\cos(x)+C3\sin(x)+Yp\]
now how do I find Yp?
should I find the derivative ?
familiar with variation of parameters ?
yes, more or less
I really need help with this one
\[y'''+y'=\frac{ \sin x }{ \cos^2x }\] 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
hmmmm not sure I'm following
I found the general solution
All I need now is the private one
Are you familiar with the method I'm using?
http://gyazo.com/85b4d43f26ed36b347d32629ee0df5b0
\[v''+v=\frac{ \sin x }{ \cos^2x }\] earlier you worked the general solution, general solution for above reduced DE is \(c_2\cos x+c_3\sin x\), yes ?
+C1
No, we're only looking at reduced DE for now.
oh ok
Can you please show how to do it using the formula you've attached?
Okay, so from general solution we have \(y_1 = \cos x\) \(y_2 = \sin x\) Wronkskian\(W(y_1, y_2) = \begin{vmatrix} \cos x &\sin x\\\cos'x&\sin'x\end{vmatrix} = \cos^2x+\sin^2x = 1 \)
simply plug it in the formula and evaluate the integral(s)
but what is g(t)?
whatit refers to?
g(t) is whatever there on the right hand side
\(g(\color{red}{x}) = \dfrac{\sin x}{\cos^2x}\)
and how do I find C1 then?
we will worry about that in the very end
oh ok
Keep in mind we're solving the "reduced" DE in \(v\)'s completely first, then we're gona substitute back \(v\)
roger that
I think I can handle with that by myself, but how then can I find C1?
btw thanks a lot!
\(\large v(x) = c_2\cos x + c_3\sin x+Y_p\) plug in \(v(x) = y'\) back
and integrate
thank you so much @ganeshie8 !
I think I've got it! I'll try at home and see if it works out for me
good question, it was useful for me as well .
you may refer to this solution generated by wolfram if you get stuck..
1 Attachment
Y_h portion is always easier than finding the Y_p
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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