# Finding if a solution exists

1. May 24, 2004

hi
this is my first attempt at using the latex commands ..
consider this Cauchy Problem:
$$u_x\exp y +u_y\exp(x)=1$$
$$u(t,tk^2)=k\exp(-t)$$
where k is a constant.
forgive me i have given up using latex as i am too slow, it would take me a week to type it up at this rate. anyway
i am trying to find the values of k where
a) this problem has a unique solution
b) no solutions

for part a) i know that it all boils down to showing where the vector (1,k^2) is never parallel to ( exp(t*k^2), exp(t) )
i eventually get to:
t cant be equal to 2ln(k)/(1-k^2), however the question asks for values of k and i havent done that. but i dont know any way to find explicit values of k

for part b) i just said t=2ln(k)/(1-k^2)

is it reasonable to give the answer in the form i have? i'm not sure
hope you can help

thanks,