show the initial value problem x(dy/dx)=4y, y(0)=1 has no solution. does this contradict the existence theorem. please explain
The Attempt at a Solution
it is easy to find out a general solution is y=C*e^(4x), C is a constant. and for any x the right part of the equation will always be
~~~~I'm sorry, I made a mistake here, the general solution should be y=C*x^4
bigger than zero, so there is no solution for y(0)=-1, the question is how to explain it?
the existence theorem I learned from class is the following:
For F(t,y,y')=0, I.C. y(x0)=y0 ...(EQ1)
Let R be the region a<x<b, c<y<d. Such that (x0,y0) belongs to R. If f(x,y) is continuous and bounded on R, then EQ1 has a solution. The validity of the solution is in R.