A Handy Method for Determining the Particular Solutions to theThird-Order Non-homogeneous Euler Equations

The general formula for determining the particular solutions is presented in this paper for a kind of third-order Euler equation likex ³( d ³y)/( dx ³)+px ²( d ²y)/( dx ²)+qx( dy)/( dx)+ry=x ^λA cos(ω ln|x|)+B sin(ω ln|x|). Two basic properties for the linear differential equation are first introduced. Then by use of the variable transformation method, the equations are transformed into the linear ordinary differential equations with constant coefficients. And then by the method of undetermined coefficient and complex number method, the simplified formula for determining the particular solution to the equations is worked out. Through some numerical examples the effectiveness and practicality of the method proposed are proven.