Given an IVP

an(x)dnydxn+an−1(x)dn−1ydxn−1+…+a1( x)dydx+a0(x)y=g(x)

y(x0)=y0, y′(x0)=y1, ⋯, y(n−1)(x0)=yn−1 If the coefficients an(x),…,a0(x) and the right hand side of the equation g(x) are continuous on an interval I and if an(x)≠0 on I then the IVP has a unique solution for the point x0∈I that exists on the whole interval I. Consider the IVP on the whole real line

sin(x)d2ydx2+cos(x)dydx+sin(x)y=tan (x)

y(0.5)=20, y′(0.5)=3, The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval