Someone Please Check Where I Went Wrong

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The discussion revolves around a Laplace Transform problem involving the equation dy/dt = -y + t^2. The user initially derived the inverse Laplace transform as y = t^2 - e^-t but found a discrepancy with the book's answer, which includes additional terms. They realized the error was in the partial fraction decomposition, needing to distribute correctly among multiple terms. The user corrected their approach, indicating they have resolved the issue. The conversation highlights the importance of accurate decomposition in solving Laplace Transform problems.
ColdFusion85
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This is a Laplace Transform problem. dy/dt = -y + t^2. I formed the inverse laplace transform equation which i verified is correct, then i took partial fraction decompositions and got for a final answer that y = (L inverse) [2/s^3 - 1/(s+1)]. and hence, y = t^2 - e^-t. however the book states that the answer is t^2 - 2*t + 2 - e^-t. where did i go wrong??
 
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Oops, I think it has to be distributed as \frac{A}{(s+1)} + \frac{(Bx+C)}{s}+ \frac{(Dx+E)}{(s^2)}+\frac{(Fx+G)}{(s^3)}, doesn't it?
 
nevermind, got it.
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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