2° order linear, homogeneous, variable coefficients

In summary, the conversation is about solving the equation sin(x) * y''(x) + 2cos(x) * y(x) = 0 with initial conditions y(0) = 0 and y'(0) = 1. The individual discussing the problem has already created a differential equation starting from a simple function and is seeking a way to find a solution without having to try by chance. They have tried various methods such as reduction of order and Fourier transformation, and have ultimately found the solution using a process called "reduction of order."
  • #1
lightarrow
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sin(x) * y''(x) + 2cos(x) * y(x) = 0
y(0) = 0
y'(0) = 1

how do I solve it?
(I know the solution because I have created the diff. equation starting from a simple function).

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lightarrow
 
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  • #2
This is equation of form: ##y'' + p(x)y' + q(x)y = g(x)## where ##p=g=0##.
Have you tried looking it up? Do you know one solution to the DE (then try reduction of order)?
 
  • #3
Simon Bridge said:
This is equation of form: ##y'' + p(x)y' + q(x)y = g(x)## where ##p=g=0##.
Have you tried looking it up? Do you know one solution to the DE (then try reduction of order)?
Yes I tried but as you suggest I've only found a text on DE where it says "...if you know two independent solutions of the equation..."
Apart from the fact that I only know one solution, which is e-x⋅sin(x), I was asking if there were a way to find such solution/s without having to try by chance (in this case I didn't because, as I wrote, I started from that function and then I found a DE that it satisfies).

I also tried with the substitution y(x) = et(x) but the new DE in t(x), that is: t''(x) + [t'(x)]2 = -2cot(x), is a Riccati equation (after substitution t'(x) = u(x) ) the resolution of which becomes again the initial DE :smile:.

I'm wondering if we could do anything by Fourier transforming the equation...

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lightarrow
 
Last edited:
  • #4
lightarrow said:
Yes I tried but as you suggest I've only found a text on DE where it says "...if you know two independent solutions of the equation..."
I correct myself: actually we only have to know one, y1(x). To find the general solution y(x) we write y(x) = y1(x)*z(x) and after substitution in the DE we find another DE in z(x) which in this case is possible to solve, but not in terms of finite combination of elementary functions, it's infact the primitive of the function e^2x / [sin(x)]^2.

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  • #5

Related to 2° order linear, homogeneous, variable coefficients

1. What is a 2nd order linear, homogeneous, variable coefficient equation?

A 2nd order linear, homogeneous, variable coefficient equation is a mathematical equation that involves a second derivative, has no constant term, and the coefficients of the equation are dependent on the independent variable.

2. How do you solve a 2nd order linear, homogeneous, variable coefficient equation?

To solve a 2nd order linear, homogeneous, variable coefficient equation, you can use the method of undetermined coefficients or variation of parameters. These methods involve finding a particular solution and using it to find the general solution of the equation.

3. What are some real-world applications of 2nd order linear, homogeneous, variable coefficient equations?

2nd order linear, homogeneous, variable coefficient equations have many applications in physics, engineering, and other scientific fields. Some examples include modeling the motion of a spring-mass system, describing the behavior of electrical circuits, and predicting the growth of populations.

4. How do variable coefficients affect the solutions of a 2nd order linear, homogeneous equation?

Variable coefficients can significantly affect the solutions of a 2nd order linear, homogeneous equation. They can change the behavior of the solutions, such as making them oscillate or grow exponentially. They can also make the solutions more complex and difficult to find.

5. Are there any special cases of 2nd order linear, homogeneous, variable coefficient equations?

Yes, there are several special cases of 2nd order linear, homogeneous, variable coefficient equations. These include equations with constant coefficients, equations with identical roots, and equations with complex roots. Each of these cases has its own unique solution method.

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