1. ## Change of variables

He guys,

I have some questions about change of variables
I have (d^2y/dx^2)*x+x=0 and want to use the substitution z=x^2
so dy/dx=(dy/dz)*(dz/dx)=2x*(dy/dx)
But is this correct (d^2y/dx^2)=4(x^2)*(d^2y/dx^2)

My second question is, how to make the substitution z=yx in the same equation?

thanks for the help

2. Originally Posted by sung
He guys,

I have some questions about change of variables
I have (d^2y/dx^2)*x+x=0 and want to use the substitution z=x^2
so dy/dx=(dy/dz)*(dz/dx)=2x*(dy/dx)
But is this correct (d^2y/dx^2)=4(x^2)*(d^2y/dx^2)

My second question is, how to make the substitution z=yx in the same equation?

thanks for the help
$\displaystyle \frac{dy}{dx}=\frac{dy}{dz}\ \frac{dz}{dx}=2x\frac{dy}{dz}$

3. thanks
but is this true
$\displaystyle$
$\displaystyle (\frac{dy}{dx})^{2}=(\frac{d^{2}y}{dx^{2}}) ?$

so

$\displaystyle$
$\displaystyle (\frac{d^{2}y}{dx^{2}}) = 4x^{2}$
$\displaystyle (\frac{d^{2}y}{dz^{2}})$

4. Originally Posted by sung
thanks
but is this true
$\displaystyle$
$\displaystyle (\frac{dy}{dx})^{2}=(\frac{d^{2}y}{dx^{2}}) ?$

so

$\displaystyle$
$\displaystyle (\frac{d^{2}y}{dx^{2}}) = 4x^{2}$
$\displaystyle (\frac{d^{2}y}{dz^{2}})$
No, there is a difference.

$\displaystyle \frac{d^2y}{dx^2}$ is the second derivative of y with respect to x.

It's not the same as the square of the first derivative.

$\displaystyle \frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\ right)=\frac{d}{dx}\left(2x\frac{dy}{dz}\right)$

Use the product rule of differentiation for this.