Find an interval centered about x=0 for which the given IVP has a unique solution:
y" + (tan x)y=e^x y(0)=1, y'(0)=0
How would I find the interval?
Let's write the DE as...
$\displaystyle \cot x\cdot y^{''} + y = e^{x}\cdot \cot x$ , $\displaystyle y(0)=1$ , $\displaystyle y^{'} (0)= 0$ (1)
Now we remember that necessary condition to have one and only one solution in an interval of the x variable is that the coefficient of the term $\displaystyle y^{''}$ doesn't vanish in that inteval, so that it exists one and only one solution of (1) for $\displaystyle -\frac{\pi}{2}<x<\frac{\pi}{2}$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$