# Solve Differential Equation: Is My Work Correct?

• bancux
In summary, the student is trying to solve a second order differential equation with variable coefficients. They are not sure how to solve it, but they have attached a pdf file with information about their problem.
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

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

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

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

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

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

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

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

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

So what is the correct solution?

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.

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

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.

betel said:
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.

#### Attachments

• pf2.pdf
18.8 KB · Views: 198

## 1. How do I know if my solution to a differential equation is correct?

The best way to check if your solution is correct is by plugging it back into the original equation and seeing if it satisfies the equation. You can also compare your solution to other known solutions or use numerical methods to approximate the solution and compare it to your solution.

## 2. What are the most common mistakes when solving a differential equation?

Some common mistakes when solving a differential equation include errors in algebraic manipulations, incorrect application of integration techniques, and forgetting to include constant of integration. It is also important to carefully consider the initial conditions and boundary conditions when solving a differential equation.

## 3. How can I check if my work is correct without solving the differential equation?

If you are unable to solve the differential equation, you can still check if your work is correct by using a graphing calculator or software to plot the solution and see if it fits the given initial or boundary conditions. You can also compare your work to known solutions or ask for feedback from a colleague or mentor.

## 4. What are some common methods for solving differential equations?

Some common methods for solving differential equations include separation of variables, integration factors, substitution methods, and series solutions. The method used will depend on the type of differential equation and its complexity.

## 5. How can I improve my skills in solving differential equations?

One of the best ways to improve your skills in solving differential equations is through practice. Work on a variety of problems and seek feedback from others. You can also attend workshops or take online courses to learn new techniques and approaches. Additionally, it is important to have a strong foundation in calculus and algebra to effectively solve differential equations.

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