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Mildred

Solving differential equations

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Hi


I am not very advanced in calculas.


I need to solve for X(t)/Y(t) when t -> infinity.


X'(t)+aX(t)+cX(t)-bY(t)=0 ....1 X(0)=0

Y'(t)+bY(t)+cX(t)=0 .....2 Y(0)=0


So I thought of deriving both equations to get:


X''(t)+aX'(t)+cX'(t)-bY'(t)=0 ...3

Y''(t)+bY'(t)+cX'(t)=0 ...4


Then substituting 2 into 3 & substituting 1 into 4


X''(t)+aX'(t)+cX'(t)-b*[bY(t)]-bcX(t)=0 ....5 Y''(t)+bY'(t)+cX(t)[a+c]+bY(t)=0 .....6


Then substituting 1 into 5 Then substituting 2 into 6


X''(t)+aX'(t)+cX'(t)+bX'(t)+baX(t)=0 Y''(t)+[a+b+c]Y'(t)+[ab+cb+b]Y(t)=0


Taking the Laplace transform Taking the Laplace transform


X(s)[s^2+s+a+c+sa+sc+sb+ba]=0 Y(s)[s^2+sa+2sb+sc+sab+scb-s-b]


Then


X(s)/Y(s)=[s^2+sa+2sb+sc+sab+scb-s-b]/[s^2+s+a+c+sa+sc+sb+ba]


This is where I get stuck as I don't know how to transform it back to get X(t)/Y(t)


Any help would be much appreciated.


Thank you

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You can't simply take the inverse transform; the quotient of the transforms isn't the transform of the quotient.

 

However, I'd recommend that you look at the equations just before you take the quotients. You have a full equation for X(s):

 

X(s)[s^2+s+a+c+sa+sc+sb+ba]=0

 

You should know that X(s) isn't 0 for all s; so that should allow you to "solve" for s.

 

I would note, however, that in this context, the Laplace transform technically shouldn't be applied.

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