MHB Initial Conditions in Laplace Transform of Second Order Differential Equations

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The discussion centers on the application of Laplace transforms to solve second-order differential equations with given initial conditions. The first equation, y' + 2y = 2(1 - e^(-2t), has an initial condition that raises questions about whether it should be Y(0)=0 or y(0)=0. The second equation, y'' - 2y' + y = t + e^t, has initial conditions y(0)=1 and y'(0)=0. Participants are encouraged to perform the Laplace Transform on both equations to clarify the initial conditions and derive solutions. The focus is on ensuring correct application of the Laplace Transform in the context of these initial conditions.
kJS
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And also:
y`+2y=2(1-e^-2t) Y(0)=0
y¨-2y`+y = t+e^t y(0)=1 and y`(0)=0

Please help me out here folks ;)
 
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It's a bit unusual to have an initial condition in the frequency domain. Are you sure the initial condition for the first DE is $Y(0)=0$? Or is it $y(0)=0$? In any case, what do you get when you Laplace Transform the entire equation? (For each DE.)
 

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