# Differential equations again

1. Jan 13, 2013

### cytochrome

1. The problem statement, all variables and given/known data
Show that sines and cosines are the solutions of the differential equation

f''(x) = (-σ^2)f(x)

What if a boundary condition is included that g(0) = 0?

2. Relevant equations
f''(x) = (-σ^2)f(x)

3. The attempt at a solution
Plugging in sin(σx) and cos(σx) yields an equality therefore the expression is true.

I'm just confused about the boundary condition.

If g(0) = 0 then only the sin(σx) works, correct?

2. Jan 13, 2013

### grzz

I am assuming that g(0) = 0 stands for f(0) = 0.

The solution f(x) = sin(σx) satisfies the condition that f(0) = 0 for all values of σ. But the solution f(x) = cos(σx) satisfies the given condition only for particular values of σ.

3. Jan 13, 2013

### SammyS

Staff Emeritus
I hope you mean f(0) = 0 .

Yes, if f(0) = 0, then only sin(σx) works.

cos(σx) does not work for that boundary condition.

@grzz,

For what value of σ will cos(σ∙0) = 0 ?

4. Jan 13, 2013

### grzz

Thanks SammyS for pointing out my mistake.

The last part of my post i.e.'But the solution f(x) = cos(σx) satisfies the given condition only for particular values of σ' is not correct.