First order homogenous DE with variable coefficient

In summary, the conversation discusses a differential equation with initial values and how to solve it using a reduction of order. The solution is provided and the conversation ends with a thank you.
  • #1
MrGandalf
30
0
I was stumped by this differential equation. The function x = x(t).

[tex]x^{\tiny\prime\prime} + \frac{1}{t}x^{\tiny\prime} = 0[/tex].

You have the initial values x(a) = 0 and x(1) = 1.

What I did was to introduce a new function u = x', so I ended up with the first order homogenous DE:
[tex]u^{\tiny\prime} + \frac{1}{t}u = 0[/tex].

This can't be that difficult, but the book which I am using does not give any details how to solve this particular problem. I only know how to solve it if I have a constant, or a function of t, on the right side.

The solution to the problem is:
[tex]x(t) = 1 - \frac{\log t}{\log a}[/tex].

Hope someone can shed some light on this. Thank ye, ye scurvy landlubber! Yarrr!
 
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  • #2
When you did a reduction of order, you arrived to a separable equation that isn't difficult to solve at all. There is no need to reduce order since the original equation can be rewritten in a simpler form. What happens if you multiply the second order ode by t?. Can you simplify it (i.e. the derivative of something)?
 
  • #3
Yes! Of course! How could I miss that? :)

Thank you for helping me!
 

FAQ: First order homogenous DE with variable coefficient

What is a first order homogeneous differential equation with variable coefficients?

A first order homogeneous differential equation with variable coefficients is a type of differential equation that involves a function and its derivatives, where the coefficients of the function are not constants but can vary with respect to the independent variable.

What is the general form of a first order homogeneous differential equation with variable coefficients?

The general form of a first order homogeneous differential equation with variable coefficients is y'(x) + p(x)y(x) = 0, where p(x) is the variable coefficient and y(x) is the unknown function.

How do you solve a first order homogeneous differential equation with variable coefficients?

To solve a first order homogeneous differential equation with variable coefficients, you can use the method of separation of variables. This involves separating the variables and integrating both sides of the equation to obtain the general solution. You can also use other methods such as the integrating factor method or the substitution method.

What are the applications of first order homogeneous differential equations with variable coefficients?

First order homogeneous differential equations with variable coefficients have many applications in physics, engineering, and other scientific fields. They can be used to model various physical phenomena such as population growth, heat transfer, and chemical reactions. They are also used in signal processing and control systems.

What are the boundary conditions for a first order homogeneous differential equation with variable coefficients?

The boundary conditions for a first order homogeneous differential equation with variable coefficients depend on the specific problem being solved. However, they typically involve specifying the value of the unknown function at certain points or the relationship between the function and its derivative at a particular point.

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