# Rewrite DE as linear equation

• March 30th 2010, 04:06 AM
HoneyPi
Rewrite DE as linear equation
Hi,

Is it possible to rewrite the equations

(a) $x'=\begin{cases}
\frac{x^{2}-1}{x-1} & x\neq1\\
2 & x=1\end{cases}
$

(b) $x'=\begin{cases}
\frac{x^{4}-1}{x^2-1} & x\neq1\\
2 & x=1\end{cases}
$

as linear equations?

Can someone give me a hint, please?

Honey $\pi$
• March 30th 2010, 05:11 AM
Prove It
Quote:

Originally Posted by HoneyPi
Hi,

Is it possible to rewrite the equations

(a) $x'=\begin{cases}
\frac{x^{2}-1}{x-1} & x\neq1\\
2 & x=1\end{cases}
$

(b) $x'=\begin{cases}
\frac{x^{4}-1}{x^2-1} & x\neq1\\
2 & x=1\end{cases}
$

as linear equations?

Can someone give me a hint, please?

Honey $\pi$

I'm hoping you can see that

$\frac{x^2 - 1}{x - 1} = \frac{(x + 1)(x - 1)}{x - 1}$

$= x + 1$.

As $x \to 1, f(x) \to 2$.

So you can rewrite this function as

$f(x) = x + 1$ for all $x$.

You should also be able to see that

$\frac{x^4 - 1}{x^2 - 1} = \frac{(x^2 + 1)(x^2 - 1)}{x^2 - 1}$

$= x^2 + 1$.

As $x \to 1, f(x) \to 2$.

So you can rewrite the function as

$f(x) = x^2 + 1$ for all $x$.

So, you can rewrite the first as a linear function, and the second as a quadratic function.