The ODEs that arise in practical applications often have non-rational coefficients. In such cases, DSolve attempts to convert the equation into one with rational coefficients using a suitable coordinate transformation.
Here is an equation that has Exp[x] as a coefficient. It is solved by transforming it to Bessel’s equation.
This equation (equation 2.437, page 507 of [K59]) has trigonometric coefficients. The solution is given in terms of elementary functions.
Here is an equation with a hyperbolic function in the coefficient of . The solution is given in terms of Legendre functions.