Solving 2nd Order ODEs w/ Variable Coefficients

In summary, there are only a few forms of second order homogeneous differential equations with variable coefficients that can be explicitly solved, such as Euler's equation. Other equations, like Bessel's equation, cannot be solved directly but can be studied using various tools to understand their characteristics. Additionally, knowing the first solution of an ODE can help in finding the second solution. However, there is no general method for solving all second order linear ODEs, and studying these equations can lead to acquiring many useful tools.
  • #1
spirally
4
0
I want to know if there is a general solution to a second order homogeneous differential equation with variable coefficients?
 
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  • #2
There are only few forms of such equations that can be explicitly solved (for example Euler's equation)

Other equations can not be solved directly, but they are important enough to get their own names and to be studied with different tools to give some characteristics of them (asymptotic behaviour, integral and differential identities, infinite sum expansion) but cannot be explicitly written (for example Bessel's Equation and its solutions the Bessel Functions)

You can always know the second solution of an ODE by just knowing the first solution, so from times to times guessing can come in handy.

In general I believe there are many problems to study that arise from second order linear ODE, and you can acquire many tools to do so along the way. But the bottom line is that there is no general method of solving them.
 
  • #3
thanks..great help
Was working with a friend on getting a general solution..
Has proven to be a real problem but we giving it more tries.
Thanks once again
 

1. What is a 2nd Order ODE with Variable Coefficients?

A 2nd Order Ordinary Differential Equation (ODE) with Variable Coefficients is a mathematical equation that involves a function and its derivatives up to the second order, where the coefficients (numbers multiplied to the function and its derivatives) are not constant, but rather vary with respect to the independent variable.

2. What are some examples of 2nd Order ODEs with Variable Coefficients?

Examples of 2nd Order ODEs with Variable Coefficients include the famous Schrödinger equation in quantum mechanics, the wave equation in physics, and the equation for the growth of a population in biology. These equations can be used to model various physical, chemical, and biological phenomena.

3. How do you solve a 2nd Order ODE with Variable Coefficients?

Solving a 2nd Order ODE with Variable Coefficients involves finding a function or a set of functions that satisfies the given equation. This can be done using various methods such as the method of undetermined coefficients, variation of parameters, or using series solutions. The specific method used depends on the type of equation and its coefficients.

4. What are the applications of solving 2nd Order ODEs with Variable Coefficients?

The applications of solving 2nd Order ODEs with Variable Coefficients are vast and diverse. They are used in many fields of science and engineering, such as physics, chemistry, biology, economics, and engineering. These equations can be used to model real-life situations and make predictions about the behavior of systems.

5. Are there any challenges in solving 2nd Order ODEs with Variable Coefficients?

Yes, there can be challenges in solving 2nd Order ODEs with Variable Coefficients, as these equations can be complex and difficult to solve analytically. Some equations may not have closed-form solutions, and numerical methods may be required to find approximate solutions. Additionally, the coefficients in these equations may change with time or other variables, making them even more challenging to solve.

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