Solve Differential Equation: Is My Work Correct?

[bancux]

Homework Statement

I need to work on a differential equation.
$$\frac{d^2T}{dx^2} - (m+n\ sin(kx))\ T = 0$$

Homework Equations

Is my work correct?

The Attempt at a Solution

$$\frac{d^2T}{dx^2} - (m+n\ sin(kx))\ T = 0$$
$$\frac{d}{dx}\left(\frac{dT}{dx} \right) = (m+n\ sin(kx))\ T$$

$$\int \frac{d}{T}\left(\frac{dT}{dx} \right) = \int (m+n\ sin(kx)) \ dx$$
$$\frac{1}{T}\left(\frac{dT}{dx} \right) = mx-\frac{n}{k}\ cos(kx)+C_1$$

$$\int \frac{dT}{T} = \int (mx-\frac{n}{k}\ cos(kx)+C_1)\ dx$$

$$ln T = \frac{m}{2}x^2-\frac{n}{k^2}sin(kx)+C_1x+C_2$$

$$T = e^{\frac{m}{2}x^2-\frac{n}{k^2}sin(kx)+C_1x}\ e^{C_2}$$

$$T = e^{\frac{m}{2}x^2-\frac{n}{k^2}sin(kx)+C_1x}\ C_2$$

 

[betel]

No, your third line is not correct. Separation of variables does not work for second order DE.

 

[bancux]

No, your third line is not correct. Separation of variables does not work for second order DE.

So what is the correct solution?

 

[betel]

Do you want me to solve your work or give you hints?

There is no obvious way to solve this equation. You could try a Fourier series expansion for T.

 

[bancux]

Well actually my question is because I am trying to solve the homogeneous part of a 2nd order inhomogeneous differential equation with variable coefficient.

Could you give some hints on the Fourier expansion, or any good reference to that?
My main goal is to solve the differential equation which I mentioned earlier.

Thanks

 

[betel]

So this DE is not some part of a homework assignment?
Because I did not find a nice analytic solution using mathematica.

The method to solve the DE depends also on what you want to do with the solution afterwards. Fouriersieries might give you a solution but you it could not be very useful.

So what is the background you are trying to solve this equation? Are m,n,k fixed or do they have to be chosen. For some values solutions might exist, for some not.

I'll try to find and explicit way to solve the equation in the meantime.

 

[bancux]

So this DE is not some part of a homework assignment?
Because I did not find a nice analytic solution using mathematica.

The method to solve the DE depends also on what you want to do with the solution afterwards. Fouriersieries might give you a solution but you it could not be very useful.

So what is the background you are trying to solve this equation? Are m,n,k fixed or do they have to be chosen. For some values solutions might exist, for some not.

I'll try to find and explicit way to solve the equation in the meantime.

It is not a homework though I do need some result on this.
I have attached a pdf file so that you can have a better view on my problem.

Thank you.