The discussion focuses on applying the existence and uniqueness theorem to the second order ordinary differential equation (ODE) $$y''+\tan(x)y=e^x$$ with initial conditions $$y(0)=1$$ and $$y'(0)=0$$. The user reformulates the ODE as $$\cos x\ y^{\ ''} + \sin x\ y = e^{x}\ \cos x$$ and identifies $$a_{2}(x) = \cos x$$, confirming that $$a_{2}(0) = 1 \ne 0$$. This verification establishes that a unique solution exists on an interval where $$\cos x$$ remains non-zero.
PREREQUISITESMathematics students, educators, and researchers focusing on differential equations, particularly those interested in the existence and uniqueness of solutions for initial value problems.
for an nth order initial value problemLet $$a_n(x),a_{n-1}(x),...,a_0(x)$$ and g(x) be continuous on an Interval I and let $$a_n(x)$$ 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.
find_the_fun said:I'm a little stuck getting started on this question. $$y''+\tan(x)y=e^x$$ with $$y(0)=1,y'(0)=0$$. I know the existence and uniqueness theorem
for an nth order initial value problem
How do I apply the theorem?