Second-order linear homogeneous differential equations are mathematical equations of form P(x) y^''+Q(x) y^'+R(x)y=0 where x is an independent variable. In general, we can solve only equations with constant coefficients, and therefore cannot solve the equations in question. On the other hand, second-order linear homogeneous differential equations with coefficients in form of analytic functions can be solved with power series.
In dissertation we discuss power series characteristics that we use for solving the equations in question. We can find a series solutions around two types of points, ordinary and singular points. Euler’s equation x^2 y^''+αxy^'+βy=0 is one of the examples where the equation has a singular point and α and β as real constants. When analysing Euler’s equation we find out that the form of the equations solution depends on zero value of characteristic equation (r)=r(r-1)+αr+β=0. There can be different or equal real zeroes or conjugated complex couple of zeroes. In the end, we analyse Bessel’s equation x^2 y^''+xy^'+(x^2-ν^2 )y=0 where ν is a constant with solutions of zero order.