# Mathematica: 2nd order PDE variable coefficients

• hasidim
In summary, Mathematica is a computer algebra system that can handle 2nd order partial differential equations (PDEs) with variable coefficients. It has a user-friendly function called DSolve that uses advanced algorithms to find analytical solutions for various types of 2nd order PDEs. These solutions are generally accurate, but it is recommended to check them using other methods. Programming knowledge is not required to use Mathematica for these types of equations, and it also has built-in functions for visualizing the solutions.
hasidim
$$a\text{''}[t]+B[t]*a'[t]-A[t]*a[t]==0$$

a[0] = 10^-9
a'[0] = 0
a[t] = ?

The coefficients A and B are variable over time. I HAVE solved (experimental and theoretical values) for the values of A and B over the time interval of interest!

I attempted to solve for a[t] using NDSolve as one normally would when coefficients are not variable over time. No luck.

Any help would be greatly appreciated!

## 1. What is Mathematica and how does it handle 2nd order PDEs with variable coefficients?

Mathematica is a computer algebra system that is commonly used by scientists, engineers, and mathematicians to solve complex mathematical problems. It has a built-in function called DSolve that can handle 2nd order partial differential equations (PDEs) with variable coefficients. This function uses advanced algorithms to find analytical solutions to these types of equations.

## 2. Can Mathematica handle any type of 2nd order PDE with variable coefficients?

Mathematica is capable of solving a wide range of 2nd order PDEs with variable coefficients, including elliptic, hyperbolic, and parabolic equations. However, there may be some cases where the equation is too complex for Mathematica to handle, or where the solution is not available in closed form.

## 3. How accurate are the solutions obtained using Mathematica for 2nd order PDEs with variable coefficients?

The accuracy of the solutions obtained using Mathematica for 2nd order PDEs with variable coefficients depends on several factors, such as the complexity of the equation, the precision set by the user, and the numerical methods used by Mathematica. In general, the solutions are very accurate, but it is always recommended to check the results using other methods or software.

## 4. Is a lot of programming knowledge required to use Mathematica for 2nd order PDEs with variable coefficients?

No, a lot of programming knowledge is not required to use Mathematica for 2nd order PDEs with variable coefficients. The DSolve function is very user-friendly and does not require any programming skills. However, having a basic understanding of the syntax and structure of Mathematica can be helpful in using the function effectively.

## 5. Can Mathematica visualize the solutions to 2nd order PDEs with variable coefficients?

Yes, Mathematica has built-in functions that can visualize the solutions to 2nd order PDEs with variable coefficients. For example, the ContourPlot3D function can be used to plot the solution as a 3D surface, while the StreamPlot function can be used to visualize the flow of a vector field solution. These visualizations can help in gaining a better understanding of the solutions obtained.

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