1. ## Higher order derivatives

please show me how to solve the following.
i have tried differentiation twice , but i cant get the same results..

2. Originally Posted by kin
please show me how to solve the following.
i have tried differentiation twice , but i cant get the same results..
First, I believe there's a typo. I think it should be $y = \frac{\sin x}{1-x^2}$ and not $y = \frac{\sin x}{(1-x)^2}$.

If you consider

$(1-x^2) y = \sin x$

then differentiating both sides wrt $x$ twice and adding the same expression gives

$
\left( (1-x^2)y\right)'' + (1-x^2)y = 0.
$
Expanding gives $(1-x^2)y'' - 4x y' - (1+x^2)y = 0$ - the first result.

If we let $u = (1-x^2)y$ then we have the hierarchy of differential equations

$
u^{(n+2)} + u^{(n)} = 0\;\;(1)
$
.

By induction we can show that

$
u^{(n)} = (1-x^2) y^{(n)} - 2n x y^{(n-1)} - n(n-1)y^{(n-2)}
$

and at $x = 0$

$
u^{(n)} = y^{(n)} - n(n-1)y^{(n-2)}.
$

Substitution into (1) leads to the second result

$
y^{(n+2)} -(n^2+3n+1) y^{(n)} - n(n-1)y^{(n-2)} = 0
$
.

3. Originally Posted by Danny
First, I believe there's a typo. I think it should be $y = \frac{\sin x}{1-x^2}$ and not $y = \frac{\sin x}{(1-x)^2}$.
this is reallly important , thanks danny....