Daniel Wood
Member
Hi guys. Im having a problem with one of my further maths assignments for college.
Unfortunately i am better with electronics than math. I hope someone would be able to provide some friendly advice on this question. it would be a great help
First i have to solve using laplace transforms
[LATEX]\frac{d^2x}{dt} -4x = 24cos(2t)[/LATEX]
given that t=0 and x = 3 and dx/dt = 4
I tried to have a dig at this myself and ended up getting stuck half way through
Firstly I written some shorthand to make solving this easier
[LATEX]\bar{x} = L \left\{ x \right\}[/LATEX]
[LATEX]x_0[/LATEX] = Value of x when t = 0
[LATEX]x_1[/LATEX] = dx/dt when t=0
So i took laplace transforms throughout and ended up with this
[LATEX]S^2 \bar{x}-Sx_0 -x_1 - 4 \bar{x} = \frac{24S}{S^2+4}[/LATEX]
Then i insert the initial condition's and rearrange the formula to get
[LATEX]\bar{x} = \frac{3S^3 +4S^2 + 36S +16}{(S^2 +4)(S-2)(S+2)} \equiv \frac {AS+B} {S^2+4} + \frac {C} {S - 2} + \frac {D} {S-2}[/LATEX]
Im quite sure it is correct up to this point. Anyway to find the value for A, B, C, D i would have to re-arrange the formula again to get
EDIT: 4S was meant to be 4S^2
[LATEX]3S^3 + 4S^2 + 36S + 16 = (AS+B)(S-2)(S+2) + C(S+2)(S^2 + 4) + D(S-2)(S^2 + 4)[/LATEX]
From that, I should be able to find the values by using S to cancel the other letters out.
S = 2 gets me a value of C = 4
S = -2 gets me a value of D = 2
Working out B does not work out so well though. If i put S = 0, the answer would not be a whole number. And from that I cant work out what A is
Maybe there is another method of solving this equation. Any suggestions would be great
Unfortunately i am better with electronics than math. I hope someone would be able to provide some friendly advice on this question. it would be a great help
First i have to solve using laplace transforms
[LATEX]\frac{d^2x}{dt} -4x = 24cos(2t)[/LATEX]
given that t=0 and x = 3 and dx/dt = 4
I tried to have a dig at this myself and ended up getting stuck half way through
Firstly I written some shorthand to make solving this easier
[LATEX]\bar{x} = L \left\{ x \right\}[/LATEX]
[LATEX]x_0[/LATEX] = Value of x when t = 0
[LATEX]x_1[/LATEX] = dx/dt when t=0
So i took laplace transforms throughout and ended up with this
[LATEX]S^2 \bar{x}-Sx_0 -x_1 - 4 \bar{x} = \frac{24S}{S^2+4}[/LATEX]
Then i insert the initial condition's and rearrange the formula to get
[LATEX]\bar{x} = \frac{3S^3 +4S^2 + 36S +16}{(S^2 +4)(S-2)(S+2)} \equiv \frac {AS+B} {S^2+4} + \frac {C} {S - 2} + \frac {D} {S-2}[/LATEX]
Im quite sure it is correct up to this point. Anyway to find the value for A, B, C, D i would have to re-arrange the formula again to get
EDIT: 4S was meant to be 4S^2
[LATEX]3S^3 + 4S^2 + 36S + 16 = (AS+B)(S-2)(S+2) + C(S+2)(S^2 + 4) + D(S-2)(S^2 + 4)[/LATEX]
From that, I should be able to find the values by using S to cancel the other letters out.
S = 2 gets me a value of C = 4
S = -2 gets me a value of D = 2
Working out B does not work out so well though. If i put S = 0, the answer would not be a whole number. And from that I cant work out what A is
Maybe there is another method of solving this equation. Any suggestions would be great
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