willmoore21
Junior Member
- Joined
- Jan 26, 2012
- Messages
- 75
\(\displaystyle y''\)\(\displaystyle + w^2y=E_{0}sin(wt)\) where w and Eo constants. (E0 is supposed to be E0 but can't get subs to work in Latex)
I need to find the general solution.
\(\displaystyle GS=PI+CF\)
First I need to find the complimentary function, I have done this by:
Try \(\displaystyle y=exp(at)\) ( where a = alpha)
\(\displaystyle a^2+w^2=0\)
2 complex distinct solutions, \(\displaystyle iw,-iw\)
Therefore CF = \(\displaystyle Aexp(iwt)+Bexp(-iwt)\)
This is one part where I am confused. I can simplify this further in the form of exp(at)((Ccos(at)+Dsin(at)), I think it is this exp(wx)((Ccos(x)+Dsin(-x)). Even if this is wrong I can continue do solve the particular integral.
So now I need the particular integral (I think this is a Wronskian as it is under out variation of parameters part of our course)
Apologies (can't get y1 and y2 subs to work in Latex)
\(\displaystyle so: PI=\)\(\displaystyle u(t)y1(t)\)\(\displaystyle +v(t)y2(t)\)
\(\displaystyle u(t)=-int(y2(t)F(t)/W(y1,y2)dt\)
\(\displaystyle v(t)=int(y2(t)F(t)/W(y1,y2)dt\)
\(\displaystyle W(y1,y2)=y1y2'-y2y1'\)
From here do I just procede as normal? I haven't come across a complex wronskian before so wasn't sure? I am happy with yes/no answers unless any of the previous is wrong. I will work out what I think it is as and when I get a reply.
I am NOT asking for a solution here, just a verification and a "go ahead" to carry on.
Thanks.
I need to find the general solution.
\(\displaystyle GS=PI+CF\)
First I need to find the complimentary function, I have done this by:
Try \(\displaystyle y=exp(at)\) ( where a = alpha)
\(\displaystyle a^2+w^2=0\)
2 complex distinct solutions, \(\displaystyle iw,-iw\)
Therefore CF = \(\displaystyle Aexp(iwt)+Bexp(-iwt)\)
This is one part where I am confused. I can simplify this further in the form of exp(at)((Ccos(at)+Dsin(at)), I think it is this exp(wx)((Ccos(x)+Dsin(-x)). Even if this is wrong I can continue do solve the particular integral.
So now I need the particular integral (I think this is a Wronskian as it is under out variation of parameters part of our course)
Apologies (can't get y1 and y2 subs to work in Latex)
\(\displaystyle so: PI=\)\(\displaystyle u(t)y1(t)\)\(\displaystyle +v(t)y2(t)\)
\(\displaystyle u(t)=-int(y2(t)F(t)/W(y1,y2)dt\)
\(\displaystyle v(t)=int(y2(t)F(t)/W(y1,y2)dt\)
\(\displaystyle W(y1,y2)=y1y2'-y2y1'\)
From here do I just procede as normal? I haven't come across a complex wronskian before so wasn't sure? I am happy with yes/no answers unless any of the previous is wrong. I will work out what I think it is as and when I get a reply.
I am NOT asking for a solution here, just a verification and a "go ahead" to carry on.
Thanks.
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