Question

Differential Equations SOS PLEASE

Answer 1

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

Answer 2

\[K^3+K=0\]

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

now how do I find Yp?

Answer 7

should I find the derivative ?

Answer 8

familiar with variation of parameters ?

Answer 9

yes, more or less

Answer 10

@ganeshie8 ?

Answer 11

I really need help with this one

Answer 12

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

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

Answer 13

hmmmm not sure I'm following

Answer 14

I found the general solution

Answer 15

All I need now is the private one

Answer 16

Are you familiar with the method I'm using?

Answer 17

http://gyazo.com/85b4d43f26ed36b347d32629ee0df5b0

Answer 18

\[v''+v=\frac{ \sin x }{ \cos^2x }\]

earlier you worked the general solution,
general solution for above reduced DE is \(c_2\cos x+c_3\sin x\), yes ?

Answer 19

+C1

Answer 20

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

Answer 21

oh ok

Answer 22

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

Answer 23

Okay, so from general solution we have
\(y_1 = \cos x\)
\(y_2 = \sin x\)

Wronkskian\(W(y_1, y_2) = \begin{vmatrix} \cos x &\sin x\\\cos'x&\sin'x\end{vmatrix} = \cos^2x+\sin^2x = 1 \)

Answer 24

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

Answer 25

but what is g(t)?

Answer 26

whatit refers to?

Answer 27

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

Answer 28

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

Answer 29

and how do I find C1 then?

Answer 30

we will worry about that in the very end

Answer 31

oh ok

Answer 32

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

Answer 33

roger that

Answer 34

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

Answer 35

btw thanks a lot!

Answer 36

\(\large v(x) = c_2\cos x + c_3\sin x+Y_p\)

plug in \(v(x) = y'\) back

Answer 37

and integrate

Answer 38

thank you so much @ganeshie8 !

Answer 39

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

Answer 40

good question, it was useful for me as well .

Answer 41

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

Answer 42

Y_h portion is always easier than finding the Y_p

Answer 43

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. >_<