Fundamental- solving a second order differential equation

[svishal03]

I've completed my Engineering but doing a self study course in Dynamics of Structures and have got a very fundamnetal question concerning solution of differential equation and hope someone will be able to help me.

Sorry if its too fundamnetal and stupid!

Let us say we have to solve a differential equation:

mu_double_dot + k_u=0

(double dot indicates second derivative)

We put,

u = e^st

we get first and second order derivative of u and substitute in the abovce differential equation and get:

s = +i sqrt(k/m) or -i sqrt(k/m)

We now say that general solution is:

u = A1 e^s1t + A2 e^s2t

where s1 and s2 are +i sqrt(k/m) and -i sqrt(k/m) respectively

Shouldn't the solution be:

u = e^s1 t OR u = e^s2t ??

Please can anyone help what is the logic in putting

u = A1 e^s1t + A2 e^s2t and why?

Vishal

 

[Mandlebra]

Linear superposition. if y1 and y2 are solutions, then y1 + y2 is also.

 

[svishal03]

Dear Mandlebra,

Thank you for the response...

Did you mean:

If y1 and y2 are solutions then :

Ay1+By2 are also solutions

where A and B are constants- did you miss put the A and B in your response?

Vishal

 

[Mandlebra]

Yes, I forgot

 

[yus310]

Yes, I forgot

You are solving the typical equation for a oscillator vibrating.

You are on the right track.

But you got imaginary roots.

what is exp^(i*root).. what is this in terms of trig functions?

Then it will make sense

good job,
yus310