# change of variables

• Jan 31st 2009, 05:37 PM
razorfever
change of variables
part (a)
use the substitution u = (y-x) to find the general family of solutions of the differential equation:

dy/dx = sin(y-x)

part (b)
show that if a new independant variable is defined by y = z^(1-n) then the differential equation:

dz/dx + p(x)z = q(x)z^n where (n not equal to 1)

becomes a linear differential equation in y(x)

part (c)
use part (b) to solve the initial value problem

xz dz/dx = x^2 + 3z^2 where z(1) = 1
• Jan 31st 2009, 05:46 PM
mr fantastic
Quote:

Originally Posted by razorfever
part (a)
use the substitution u = (y-x) to find the general family of solutions of the differential equation:

dy/dx = sin(y-x)

[snip]

I only have time to give you a start for part (a):

$\displaystyle \frac{du}{dx} = \frac{dy}{dx} - 1 \Rightarrow \frac{dy}{dx} = \frac{du}{dx} + 1$. So the DE is:

$\displaystyle \frac{du}{dx} + 1 = \sin u$.

Therefore $\displaystyle x = \int \frac{1}{\sin (u) - 1} \, du$.

Solve for x in terms of u and then substitute back $\displaystyle u = y - x$.