Voofie | Worked examples of solving linear non-homogeneous ordinary differential equation by operator factorization

Worked examples of solving linear non-homogeneous ordinary differential equation by operator factorization

25 Jun 10

This tutorial provides several example in solving linear non-homogeneous ordinary differential equation(NVODE) using operator factorization method.

Content

1 Introduction
2 Examples
- 2.1 Example 1:

Introduction

This tutorial provides several examples in solving linear non-homogeneous ordinary differential equation(NVODE) using operator factorization method^[1].

Examples

Example 1:

To solve:

$\f$f''(x)+\frac{1}{x(1- ln x)}f'(x)+\frac{1}{x^2 (ln x-1)}f(x)=\frac{1}{x^2}\f$$ .........................(1)

To solve 1 using operator factorization method, we first need to find a fundamental solution to the equation:

$\f$L y_1 = 0\f$$ where $\f$L = D^2+\frac{1}{x(1- ln x)}D+\frac{1}{x^2 (ln x-1)}\f$$

For the simplest guess, we may try constant or polynomial of x. We notice that if $\f$y_1 = x\f$$ ,

$\f$L x = 0\f$$

Therefore, we know that x is a fundamental solution to L.

Now, we can find $\f$b_1\f$$ :

$\f$b_1 = -\frac{y_1'}{y_1}\f$$

$\f$b_1 = -\frac{1}{x}\f$$

We know that L can be factorized into:

$\f$L = (D+h_0) (D+b_1)\f$$

From this equation:

$\f$\begin{cases} h_{n-1} = a_n & & h_{j-1}=a_j-\sum _{k=j}^{n-1} h_kC_j^kb_1^{(k-j)}& \text{ if } j=1,...,n-1 & \end{cases}\f$$

We have:

$\f$h_0 = a_1 - h_1 b_1\f$$

$\f$h_0 = \frac{1}{x(1- \ln x)} + \frac{1}{x}\f$$ since $\f$h_1 = 1\f$$

So, we have factorized L totally into:

$\f$L = \left(D+\frac{1}{x}+\frac{1}{x(1- \ln x)}\right)\left(D-\frac{1}{x}\right)\f$$

Therefore, putting it into (1), we have:

$\f$L y = \frac{1}{x^2}\f$$

$\f$\Rightarrow y = \left(D-\frac{1}{x}\right)^{-1}\left(D+\frac{1}{x}+\frac{1}{x(1- \ln x)}\right)^{-1} \frac{1}{x^2}\f$$

$\f$\Rightarrow y=\left(D-\frac{1}{x}\right)^{-1}e^{-\int \left(\frac{1}{x}+\frac{1}{x(1- \ln x)}\right) \, dx}\left(\int e^{\int \left(\frac{1}{x}+\frac{1}{x(1- \ln x)}\right) \, dx}\left(\frac{1}{x^2}\right)dx+C_1\right)\f$$

$\f$\Rightarrow y=\left(D-\frac{1}{x}\right)^{-1}\left(\frac{\ln x -1}{x}\right)\left(\ln (\ln x-1)+C_1\right)\f$$

$\f$\Rightarrow y=e^{\int \frac{1}{x} \, dx}\left(\int e^{-\int \frac{1}{x} \, dx}\left(\frac{\ln x -1}{x}\right)\left(\ln (\ln x-1)+C_1\right)dx+C_2\right)\f$$

$\f$\Rightarrow y=x\int \left(\frac{\ln x -1}{x^2}\right)\left(\ln (\ln x-1)+C_1\right)dx+C_2x\f$$

$\f$\Rightarrow y=x\int \left(\frac{\ln x -1}{x^2}\right)\ln (\ln x-1)dx-C_1\ln x+C_2x\f$$

You can put it back into (1) to verify that it is indeed a general solution since it contains 2 arbitrary constants.

Footnotes and Citations

Ross Tang" Solving linear non-homogeneous ordinary differential equation with variable coefficients with operator method",Retrieved 25 June 2010

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Worked examples of solving linear non-homogeneous ordinary differential equation by operator factorization

Introduction

Examples

Example 1:

Footnotes and Citations

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