Thread: Making a substitution in a differential equation.

1. Making a substitution in a differential equation.

The variables x and y are related by the differential equation:

$\displaystyle \frac{dy}{dx} = \frac{x^2 - y^2}{xy}$

Use the substitution y=xz, where z is a function of x, to obtain the differential equation:

$\displaystyle x\frac{dz}{dx} = \frac{1-az^2}{z}$

My questions sheet has 2 instead of the a, but I get a=1, which is correct?

Cheers.

2. Originally Posted by Nyoxis
The variables x and y are related by the differential equation:

$\displaystyle \frac{dy}{dx} = \frac{x^2 - y^2}{xy}$

Use the substitution y=xz, where z is a function of x, to obtain the differential equation:

$\displaystyle x\frac{dz}{dx} = \frac{1-az^2}{z}$

My questions sheet has 2 instead of the a, but I get a=1, which is correct?

Cheers.
The a should be a 2.

$\displaystyle x \frac{dz}{dx} + z = \frac{x^2 - x^2z^2}{x^2z}$

and isloating z' gives

$\displaystyle x \frac{dz}{dx} = \frac{1 - 2z^2}{z}$.

3. Originally Posted by danny arrigo
The a should be a 2.

$\displaystyle x \frac{dz}{dx} + z = \frac{x^2 - x^2z^2}{x^2z}$
Ah I see where I went wrong, I forgot to use the product rule when differentiating y=xz, thanks for the help.