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Second Order Differential Equations question

Toast210

The angular displacement from its equilibrium position of a swing door is modelled by the differential equation:
(d²θ/dt²) + 4(dθ /dt) + 5θ = 0

Boundary condition : θ = π/4 when t = 0

How do I find the particular solution? Struggling with the constants.

Edit 1: Was thinking of differentiating to get velocity and make that 0 but not too sure

(edited 3 years ago)

Reply 1

ran-dumb

Solve the differential equation, ignoring the boundary condition first. What have you tried?

Reply 2

mqb2766

Original post by Toast210

You realise the rhs is zero so the differential equation is homogeneous so the particular solution/integral is zero. Unless I've misunderstood something?

(edited 3 years ago)

Reply 3

Toast210

Original post by mqb2766

You realise the rhs is zero so the differential equation is homogeneous so the particular solution/integral is zero. Unless I've misunderstood something?

The particular solution and integral are different to my knowledge. I'm not sure the best way to describe it other than the particular solution is finding constants of the equation using the boundary conditions.

Reply 4

Toast210

Original post by *****deadness

Solve the differential equation, ignoring the boundary condition first. What have you tried?

So I got the equation y = e^-2t(Acost + Bsint)

Using the first boundary condition, I got A as π/4 however I'm not sure about finding B

Reply 5

mqb2766

Original post by Toast210

In that case you really need two boundary values. You only have one so will have one degree of freedom remaining. You could assume d theta/dt is zero at t=0, but the question should clearly state that.

(edited 3 years ago)

Reply 6

Toast210

Original post by mqb2766

See I was thinking that because displacement is 0, velocity is also 0. However, I wasn't too sure and wanted to clarify. But it seems that's the only way I could do it right?

Reply 7

mqb2766

Original post by Toast210

See I was thinking that because displacement is 0, velocity is also 0. However, I wasn't too sure and wanted to clarify. But it seems that's the only way I could do it right?

You can't necessarily assume that but it is a common occurrence with mechanical systems. With a 2nd order differential equation you need two initial / boundary conditions to solve it uniquely. The question should clearly state it. Where does it come from?

Reply 8

mqb2766

http://reader.epubee.com/books/mobile/5a/5a05d67dc55c6fbe14b9bf21818fcfeb/text00187.html
The door starts from rest so d theta/dt=0 when t=0.

(edited 3 years ago)

Reply 9

DFranklin

Original post by Toast210

See I was thinking that because displacement is 0, velocity is also 0. However, I wasn't too sure and wanted to clarify. But it seems that's the only way I could do it right?

Displacement isn't 0 - it's

\pi/4

Original post by mqb2766

http://reader.epubee.com/books/mobile/5a/5a05d67dc55c6fbe14b9bf21818fcfeb/text00187.html
The door starts from rest so d theta/dt=0 when t=0.

Kind of depressing that skill in finding the actual question the OP has posted is so important on the forum these days, although I salute you for it! (PRSOM, sigh).

Reply 10

mqb2766

Original post by DFranklin

Kind of depressing that skill in finding the actual question the OP has posted is so important on the forum these days, although I salute you for it! (PRSOM, sigh).

I should have realised that the wording after the ode wasn't from the original question, but yes, it's always good to have a pic of the full question.

Reply 11

Annabelle.1645

The solution has complex roots and these can be found by solving using the quadratic formula and substituting in using i=root(-1). Now you have two roots to the auxiliary equation and can form a general solution in the form theta= Acos(t)e^(-ct) Bsin(t)e^(-ct). Then you know theta(0)=pi/4, and theta'=0- so substituting in these values into the general solution and it's derivative the particular solution can be found. I get A=pi/4 and B=pi/2. Hope this helps :smile: