2° order linear, homogeneous, variable coefficients

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Discussion Overview

The discussion revolves around solving a second-order linear homogeneous differential equation with variable coefficients, specifically the equation sin(x) * y''(x) + 2cos(x) * y(x) = 0, along with initial conditions y(0) = 0 and y'(0) = 1. Participants explore methods for finding solutions, including reduction of order and potential transformations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant presents the differential equation and asks for a solution, noting they derived it from a simple function.
  • Another participant identifies the equation's form and suggests looking up solutions or using reduction of order if one solution is known.
  • A participant acknowledges knowing one solution, e^(-x) * sin(x), but expresses difficulty in finding a second independent solution without resorting to chance.
  • The same participant discusses attempting a substitution that leads to a Riccati equation, which ultimately returns to the original differential equation.
  • Further clarification is provided that knowing one solution allows for the construction of the general solution through a specific method, although the resulting integral does not yield elementary functions.
  • Another participant confirms the method of reduction of order and provides a link for further reference, while noting they have not verified the previous participant's solution.

Areas of Agreement / Disagreement

Participants generally agree on the method of reduction of order and the need for a second solution, but there is no consensus on how to find that second solution or on the effectiveness of the proposed methods.

Contextual Notes

The discussion includes assumptions about the existence of solutions and the applicability of methods like reduction of order and Fourier transforms, which remain unresolved in terms of their effectiveness for this specific equation.

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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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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)?
 
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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Last edited:
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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