Differential equations y''-xy'+2y=0?

In summary, the conversation discusses how to solve a linear homogeneous differential equation with variable coefficients using power series. The given solution is in the form of a series, and the steps to solve the equation involve calculating derivatives, plugging into the equation, and finding a recurrence relation. The conversation also briefly mentions two types of power series, Σ anxn and λxΣ anxn, and when to use them.
  • #1
hawaiidude
41
0
how would you solve

y''-xy'+2y=0?

it says, {2a2+6a3x+12a4x^2+...n(n-1_anx^n-1

ahh i don't know..how would you solve this? can someone explain? my math teacher is un willing to help me...
 
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  • #2
You mean your teacher is unwilling to tell you the answer. How mean of him! Does he expect you to actually LEARN how to do it?

The problem you have is a linear homogeneous d.e. with variable coefficients. The standard way of solving such an equation is to use power series (as you can see from the given solution).

Let y= Σ anxn where the an are to be determined. Calculate the derivatives of that, plug into your equation and combine like powers of x. You will get a recurrence relation for the ans.
 
  • #3
yeah..thnaks but why are there two types of reserection and when do i use them? like sigma n=0 anX^n and other one with the & lambda ; x^ sigma anX^n...when do i use which ones and how? and how do i plug in answers in the differential equation/ like 8x^2y''+10xy'+(x-1)y=0
x^ &lambda ; {8 &lambda ; -1) a0 +10 &lambda ; ...and so on...any help is welcomed
 
  • #4
Please do not post the same thing on different threads. I responded to this on a different thread.
 

1. What is a differential equation?

A differential equation is a mathematical equation that relates an unknown function to its derivatives. It describes how a quantity changes over time or space.

2. What does the notation y''-xy'+2y=0 mean?

The notation y''-xy'+2y=0 is a second-order linear differential equation, which means it involves the second derivative (y'') of the unknown function y. The terms involving y' and y are multiplied by the variable x and the constant 2, respectively.

3. How do you solve a differential equation?

There are various methods for solving differential equations, depending on their type and complexity. Some common techniques include separation of variables, substitution, and using an integrating factor. In general, it involves finding a function that satisfies the given equation.

4. What is the significance of the solution to a differential equation?

The solution to a differential equation represents the behavior of the unknown function over time or space. It can provide valuable insights into the system being modeled and can be used to make predictions and solve real-world problems.

5. How are differential equations used in science and engineering?

Differential equations are used in many fields of science and engineering to model and understand various natural phenomena and processes. They are particularly useful in physics, chemistry, biology, and engineering disciplines such as electrical, mechanical, and chemical engineering.

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