Particular differnetial solution

  • Thread starter Phymath
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In summary, to solve for the particular solution of the damped harmonic oscillator driven by the damped harmonic force, you can use the hint given and find the solution in the form x(t) = De^{Bt - i\phi}. By substituting this into the equation and solving for the coefficients, you can find the particular solution. This requires a lot of work, but it will ultimately lead to the correct solution.
  • #1
Phymath
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solve for the particular solution of the damped harmonic oscillator driven by the damped harmonic force
[tex] F(t) = F_0e^{-\alpha t)cos(\omega t)[/tex]

(Hint: [tex]e^{-\alpha t} cos(\omega t) = Re[e^{-\alpha t + i \omega t}] = Re[e^{B t}][/tex] where [tex]B = -\alpha + i \omega[/tex]. Find the solution in the form [tex]x(t) = De^{B t - i \phi} [/tex], i don't have much diff e q the only thing i think of doing is the following..

[tex]x'' + 2 \gamma x' + \omega^2 x = 0 [/tex]
[tex] c^2 + 2 \gamma c + \omega^2 = 0 [/tex]
[tex] x(t) = C_1 e^{-(\sqrt{\gamma^2 - \omega^2}+\gamma)t} + C_2 e^{(\sqrt{\gamma^2 -\omega^2}-\gamma) t}[/tex]
no idea where to go from here... any help would be awesome
 
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  • #2
I forget what the damped harmonic oscillator DE is but it looks like you solved the homogenous part of the equation. Now all you need is a single solution to the inhomogenous equation.

To do this, follow the hint they've given you, take a wild chance and just substitute [itex]Re[e^{Bt}][/itex] for x. You won't get the right answer (that is, get F(t) on the RHS of your DE), but you should get something that is proportional and that will give you enough hint to get you through.

Good luck. It's a lot of work. Feel proud. Time for me to get some sleep.

Carl
 

FAQ: Particular differnetial solution

What is a particular differential solution?

A particular differential solution is a specific solution to a differential equation that satisfies all of the given initial conditions.

How is a particular differential solution different from a general solution?

A general solution contains all possible solutions to a differential equation, while a particular differential solution is a specific solution that satisfies given initial conditions.

How do you find a particular differential solution?

To find a particular differential solution, you must first solve the differential equation using any applicable methods such as separation of variables or variation of parameters. Then, plug in the given initial conditions to determine the specific solution.

Can a particular differential solution exist without a general solution?

Yes, a particular differential solution can exist without a general solution. This typically occurs when the initial conditions are given and a specific solution is needed, rather than a general solution that encompasses all possibilities.

How is a particular differential solution used in real-world applications?

A particular differential solution can be used to model and predict real-world phenomena, such as population growth or radioactive decay. It allows for the determination of a specific outcome based on given initial conditions, making it a useful tool in various scientific fields.

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