Differential Equation Resolve for f(x)

[Tosh5457]

Homework Statement

Resolve for f(x).

\frac{d^{2}f}{dx^{2}} = sinf

Homework Equations

The Attempt at a Solution

I haven't studied differential equations yet, so I have no idea what to do...

 

[Myung]

The derivative of Sinx = Cosx
and the The derivative of Cosx = - Sinx
df^2/ dx^2 means you derived it twice so
after deriving for sin you will derive for cos

 

[LCKurtz]

Homework Statement

Resolve for f(x).

\frac{d^{2}f}{dx^{2}} = sinf

I haven't studied differential equations yet, so I have no idea what to do...

That is a nonlinear equation which I doubt has a simple closed form solution. Sometimes such equations are linearized as an approximation. For small values of f you could use the approximation sin(f) ≈ f giving f'' - f = 0.

 

[Mark44]

The derivative of Sinx = Cosx
and the The derivative of Cosx = - Sinx
df^2/ dx^2 means you derived it twice so
after deriving for sin you will derive for cos

If f(x) = sin(x), then d²f/dx² = -sin(x), but this is very different from the problem in the original post. In that problem, the right side of the equation is sin(f), not sin(x). See LCKurtz's post.

 

[Tosh5457]

That is a nonlinear equation which I doubt has a simple closed form solution. Sometimes such equations are linearized as an approximation. For small values of f you could use the approximation sin(f) ≈ f giving f'' - f = 0.

Yea I thought of that, but f might not be small so I need more terms on taylor's series. If I go to the 3rd term in Taylor's series:

Does it have a closed form solution this way?

 

[Mark44]

I don't think so. Adding that cubic term makes the equation nonlinear.

 

[LCKurtz]

Yea I thought of that, but f might not be small so I need more terms on taylor's series. If I go to the 3rd term in Taylor's series:

Does it have a closed form solution this way?

Maple gives a solution in terms of JacobiSN functions. These involve inverses of elliptic integrals and doubly periodic functions, for what it's worth.

 

[jackmell]

Homework Statement

Resolve for f(x).

\frac{d^{2}f}{dx^{2}} = sinf

Homework Equations

The Attempt at a Solution

I haven't studied differential equations yet, so I have no idea what to do...

That equation is analogous to the non-linear pendulum which can be solved analytically. First start by integrating it:

\int \frac{d^2f}{dx^2}=\int \sin(f)

1/2 \left(\frac{df}{dx}\right)^2=-\cos(f)+c

we're then left with a first order ODE, the solution of which can be expressed in terms of elliptic functions.