Finding interval where second order ODE has unique solution

[find_the_fun]

I'm a little stuck getting started on this question. \(\displaystyle y''+\tan(x)y=e^x\) with \(\displaystyle y(0)=1,y'(0)=0\). I know the existence and uniqueness theorem

Let \(\displaystyle a_n(x),a_{n-1}(x),...,a_0(x)\) and g(x) be continuous on an Interval I and let \(\displaystyle a_n(x)\) not be 0 for every x in this interval. If x=x_0 is any point in this interval, then a solution Y(x) of the initial value problem exists on the interval and is unique.

for an nth order initial value problem

How do I apply the theorem?

 

[chisigma]

I'm a little stuck getting started on this question. \(\displaystyle y''+\tan(x)y=e^x\) with \(\displaystyle y(0)=1,y'(0)=0\). I know the existence and uniqueness theorem
for an nth order initial value problem

How do I apply the theorem?

Let's write the second order linear ODE as...

$\displaystyle \cos x\ y^{\ ''} + \sin x\ y = e^{x}\ \cos x, \ y(0)=1,\ y^{\ '} (0)=0\ (1)$

Here $a_{2}(x) = \cos x$, and is $a_{2} (0) = 1 \ne 0$, so that the existence and uniqueness theorem is verified...

Kind regards

$\chi$ $\sigma$