2nd Order ODE,Homogeneous,Non-Constant Coeff.

  • Thread starter tim85ruhruniv
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In summary, the conversation is about finding the solution for the given differential equation using ln(x) and reducing the order to find the general solution. The boundary conditions and relation between the constants c and k are also mentioned. After some trial and error, the solution is found to be y=ln(c_o+xk).
  • #1
tim85ruhruniv
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Hi,

I am trying to guess the solution for this i am sure the solution involves a ln(x) so that i can reduce the order to find the general solution but i just can't seem to find it... any suggestions ??

[tex]\[
(c_{o}+xk)y''+ky'=0\][/tex]

with ths usual boundary conditions
[tex]\[y\left(0\right)=y_{0}\qquad y\left(l\right)=y_{l}\]
[/tex]

c and k are constants and they are related
here is the relation if it is of any additional help...
[tex]\[
\frac{c_{l}-c_{o}}{l}=k\][/tex]

thanks a lot :)
 
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  • #2
phew after some hits and misses got it finally :)

[tex]\[
y=ln\left(c_{o}+xk\right)\][/tex]

seems to solve the equation...
 

1. What is a second order ODE?

A second order ODE (ordinary differential equation) is a mathematical equation that relates a function to its derivatives. It involves a dependent variable, its first and second derivatives, and an independent variable.

2. What does it mean for an ODE to be homogeneous?

An ODE is homogeneous if all of its terms have the same degree of the dependent variable and its derivatives. In other words, the coefficients of the terms do not depend on the independent variable.

3. What are non-constant coefficients in an ODE?

Non-constant coefficients in an ODE refer to the coefficients that are not constants, but may depend on the independent variable. These coefficients can affect the behavior and solutions of the ODE.

4. How do you solve a second order homogeneous ODE with non-constant coefficients?

The general method for solving a second order homogeneous ODE with non-constant coefficients is to use the method of undetermined coefficients. This involves finding a particular solution based on the form of the coefficients and then combining it with the complementary solution to get the general solution.

5. What are some real-life applications of second order homogeneous ODEs with non-constant coefficients?

Second order homogeneous ODEs with non-constant coefficients are commonly used in physics, engineering, and other fields to model various phenomena such as oscillations, vibrations, and fluid flow. They can also be used to describe population growth, chemical reactions, and electrical circuits.

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