Graduate Applying the Laplace transform to solve Differential equations

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The discussion centers on the application of the Laplace transform to differential equations, specifically questioning whether it can transform a finite-order equation into one of infinite order. The Laplace transform simplifies differential equations into algebraic equations, allowing for easier manipulation. An example provided is the equation y''(t) + sin(t)y(t) = 0, with initial conditions y(0) = A and y'(0) = B. The participants express uncertainty about the validity and purpose of expanding terms like sin(t) into a series for the transform. Ultimately, the conversation highlights the complexities and considerations involved in applying the Laplace transform to certain types of differential equations.
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Is it possible to apply Laplace transform to some equation of finite order, second for instance, and get the differential equation of infinite order?
 
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A laplace transform turns a differential equation into an algebraic equation.
 
If you had a term like ##\sin t\,y##, you could expand ##\sin t## as a series and take the Laplace transform of the result term by term, which would give you a bunch of derivatives of Y(s). I'm not sure why you'd want to do that though or if doing so is valid.
 
pasmith said:
A laplace transform turns a differential equation into an algebraic equation.
It is only in the case when you have a differential equation with constant coefficients.
 
Do you have an example in mind?
 
##y''(t)+\sin(t)y(t)=0##, where ##y(0)=A##, ##y'(0)=B##. If I apply the Laplace transform would I get the differential equation of infinite order?
 

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