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faiz4000
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Homework Statement
The coordinates ##(x,y)## of a particle moving along a plane curve at any time t, are given by
[tex]\frac{dy}{dt} + 2x=\sin 2t,[/tex]
[tex]\frac{dx}{dt} - 2y=\cos 2t.[/tex]
If at ##t=0##, ##x=1## and ##y=0##, using Lapace transform show that the particle moves along the curve
[tex]4x^2+4xy+5y^2=4[/tex]
Homework Equations
note: Lowercase letters ##x##,##y## are functions of ##t##. Uppercase letters ##X##,##Y## are functions of ##s##.
The Attempt at a Solution
Apply Laplace Transform on the given equations
[tex]\frac{dy}{dt} + 2x=\sin 2t~~~~~~~~(1)[/tex]
Applying LT,
[tex]sY-y(0)+2X=\frac{2}{s^2+4}[/tex]
[tex]2X+sY= \frac{2}{s^2+4}~~~~~~~~(2)[/tex]
[tex]\frac{dx}{dt} - 2y=\cos 2t~~~~~~~~(3)[/tex]
Applying LT,
[tex]sX-2Y=\frac{s^2+s+4}{s^2+4}~~~~~~~~(4)[/tex]
Now Solving ##(2)## and ##(4)## simultaneously,
[tex]X=-s^3-s^2-4s-4[/tex]
[tex]Y=2s^2+8[/tex]
Now I have to apply Inverse Laplace transform to get back ##x## and ##y## but i don't know how to get ILT of a constant... Also not sure everything I've done till now is right so please help
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