Equivalence of Two Forms of Solutions to Second Order ODEs

[nickmai123]

Homework Statement

I was curious if anyone could help me prove the equivalence between the two forms of solutions to second order ODEs, one being the linear combination of two solutions and the other being the phase-shifted sin/cos function.

Homework Equations

$$\frac{d^{2}x}{dt^{2}}+\frac{k}{m}x=0$$

$$x(t)=Asin(\omega t+\phi)$$

$$x(t)=C_{1}cos(\sqrt{\frac{k}{m}}t)+C_{2}sin(\sqrt{\frac{k}{m}}t$$

$$A$$ and $$\phi$$ should be treated as constants. I know that this is the differential equation for a harmonic oscillator, but I figured since it's more of how the mathematics behind it work, it belongs in this forum. If the moderator believes it should be moved, then by all means please move it.

The Attempt at a Solution

I have no idea where to start the proof for this. The second solution I wrote can be simply verified from the characteristic equation for the differential equation.

Thanks.

EDIT: I found my error. Oops. I wrote it as r^2 + (k/m)r. Thats what I get for doing math at 3AM, lol.

 

[Pengwuino]

You're missing the second exponential in the 2nd equation.

 

[nickmai123]

You're missing the second exponential in the 2nd equation.

Hm? Which exponential? I got roots of 0 and -k/m for that differential equation's characteristic equation. I did forget the negative sign to the -k/m though.

 

[Pengwuino]

Check the roots again, the characteristic equation should be $$\lambda ^2 + \frac{k}{m} = 0$$. Assuming k>0, you have imaginary roots giving a positive and negative exponential solution.

 

[bartek2009]

I don't think the second solution satisfies the equation.

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[HallsofIvy]

Homework Statement

I was curious if anyone could help me prove the equivalence between the two forms of solutions to second order ODEs, one being the linear combination of two solutions and the other being the phase-shifted sin/cos function.

Homework Equations

$$\frac{d^{2}x}{dt^{2}}+\frac{k}{m}x=0$$

$$x(t)=Asin(\omega t+\phi)$$

$$x(t)=C_{1}+C_{2}e^{-\frac{k}{m}t}$$

$$A$$ and $$\phi$$ should be treated as constants. I know that this is the differential equation for a harmonic oscillator, but I figured since it's more of how the mathematics behind it work, it belongs in this forum. If the moderator believes it should be moved, then by all means please move it.

The Attempt at a Solution

I have no idea where to start the proof for this. The second solution I wrote can be simply verified from the characteristic equation for the differential equation.

Thanks.

Hm? Which exponential? I got roots of 0 and -k/m for that differential equation's characteristic equation. I did forget the negative sign to the -k/m though.

Then you've either got the wrong characteristic equation or you have solved it incorrectly. Neither 0 nor -k/m satisfy the characteristic equation for this d.e.

Please show your work.

 

[nickmai123]

Then you've either got the wrong characteristic equation or you have solved it incorrectly. Neither 0 nor -k/m satisfy the characteristic equation for this d.e.

Please show your work.

Got it. Thanks.

 

[HallsofIvy]

Oh, that's not fair! Show your work so we can all laugh at your mistake and feel superior!

(Some silly little arithmetic or algebra mistake.)

 

[nickmai123]

Oh, that's not fair! Show your work so we can all laugh at your mistake and feel superior!

(Some silly little arithmetic or algebra mistake.)

Haha no i just added an r for the term (k/m)x. I took DE my sr year in high school, so most of it's begun to leave me. I have a feeling it's going to come back bite me in the butt when I take circuits... sigh.