1. ## Differential equations

Hey
I am trying to calculate the streamline of u=(ay, -ax, 0)
I know i need to solve this set of differential equations:
dx/ds = ay, dy/ds= -ax , dz/ds=0

I am supposed to get a solution of:
x = yo sin (as) + xo cos (as)
y = yo cos (as) - xo sin (as)
z = zo

where xo, yo, zo are the intial conditions.
I don't understand where the sin and cos have come from.
Thank you for any help

2. Originally Posted by capricorn10
Hey
I am trying to calculate the streamline of u=(ay, -ax, 0)
I know i need to solve this set of differential equations:
dx/ds = ay, dy/ds= -ax , dz/ds=0

I am supposed to get a solution of:
x = yo sin (as) + xo cos (as)
y = yo cos (as) - xo sin (as)
z = zo

where xo, yo, zo are the intial conditions.
I don't understand where the sin and cos have come from.
Thank you for any help
from $\displaystyle \frac{1}{a} \frac{dx}{ds} = y$ and $\displaystyle \frac{dy}{ds} = -ax$

we have

$\displaystyle \frac{d^2x}{ds^2} = -a^2x$

which have solution

$\displaystyle y = c_1 \cos (as) + c_2 \sin (as)$

do the rest

-regards

3. By the way, to find just the streamlines (without knowing when you are at a particular point on the streamlines), you can eliminate the variable s. Since $\displaystyle \frac{dx}{ds}= ay$ and $\displaystyle \frac{dy}{dx}= -ax$, $\displaystyle \frac{dy}{dx}= \frac{\frac{dy}{ds}}{\frac{dx}{ds}}$$\displaystyle = \frac{-ax}{ay}= -\frac{x}{y}$. That is a separable equation: $\displaystyle ydy= -xdx$ so $\displaystyle \frac{1}{2}y^2= -\frac{1}{2}x^2+ C$ or $\displaystyle x^2+ y^2= 2C$. Fortunately, $\displaystyle \frac{dz}{ds}= 0$ so z is a constant. The streamlines are circles, in the plane $\displaystyle z= z_0$, with center at $\displaystyle (0, 0, z_0)$. You should be able to see that this is exactly what x = yo sin (as) + xo cos (as)
y = yo cos (as) - xo sin (as)
z = zo

considered as parametric equations in the parameter s, give.