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Homework Statement
Given
[itex]\frac{dx}{dt} + ax = Asin(ωt), x(0) = b[/itex]
Solve for [itex]x(t)[/itex]
Homework Equations
The Attempt at a Solution
I take the Laplace transform of both sides and get
[itex]sX(s) - x(0) + aX(s) = \frac{Aω}{s^{2} + ω^{2}}[/itex]
[itex]X(s) = \frac{b}{s + a} + \frac{Aω}{(s^{2} + ω^{2})(s+1)}[/itex]
The solution then state that the expression above is equal to
[itex](b + \frac{ωA}{a^{2}+ω^{2}})\frac{1}{s + a} + \frac{aA}{a^{2} + ω^{2}}\frac{ω}{s^{2} + ω^{2}} - \frac{ωA}{a^{2} + ω^{2}}\frac{s}{s^{2} + ω^{2}}[/itex]
I don't know how the two expressions are equal to each other or were [itex]\frac{aA}{a^{2} + ω^{2}}\frac{ω}{s^{2} + ω^{2}} - \frac{ωA}{a^{2} + ω^{2}}\frac{s}{s^{2} + ω^{2}}[/itex] came from. I believe that [itex]\frac{b}{s + a} + \frac{Aω}{(s^{2} + ω^{2})(s+a)} = (b + \frac{ωA}{a^{2}+ω^{2}})\frac{1}{s + a}[/itex] as it's just factoring. So then if this is true than [itex]\frac{aA}{a^{2} + ω^{2}}\frac{ω}{s^{2} + ω^{2}} = \frac{ωA}{a^{2} + ω^{2}}\frac{s}{s^{2} + ω^{2}}[/itex], which I don't see how that's true.
Thanks for any help.