Discuss the application of the existence and uniqueness theorem to the initial value problem
y' = (x - y)/(x + 3y) ; y(0) = -1
The equation is written in 'normal fashion'...
$\displaystyle y^{'}= f(x,y)$ , $\displaystyle y(x_{0})=y_{0}$ (1)
... where...
$\displaystyle f(x,y)= \frac{x-y}{x+3\cdot y}$ , $\displaystyle x_{0}=0$ , $\displaystyle y_{0}= -1$
The partial derivative of $\displaystyle f(*,*)$ respect to y is...
$\displaystyle f_{y}^{'}= - \frac {4\cdot x}{(x+3y)^{2}}$
Both $\displaystyle f(*,*)$ and $\displaystyle f_{y}^{'}(*,*)$ are continous in $\displaystyle [0,1]$ so that does exist one and only one solution of (1) crossing that point...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$