Answer
$y_{p_1}(x)=A e^{3x}; y_{p_2}(x)=x(Bx+C) \cos 2x+x(Dx+E) \sin 2 x$
There is no term of $y_{p_2}(x)=x(Bx+C) \cos 2x+x(Dx+E) \sin 2 x$ is a solution of the complimentary equation or auxiliary equation.
Work Step by Step
Consider $G(x)=e^{\alpha x} A(x) \sin mx $ or $G(x)=e^{\beta x} A(x) \cos mx $
The trial solution for the method of undetermined coefficients is defined as:
$y_p(x)=e^{\alpha x} B(x) \sin mx +e^{\beta x} C(x) \cos mx$
On comparing the above equation with the given equation, we get
$m=k=1$
Hence, the trial solution for the method of undetermined coefficients is as follows:
$y_{p_1}(x)=A e^{3x}; y_{p_2}(x)=x(Bx+C) \cos 2x+x(Dx+E) \sin 2 x$
There is no term of $y_{p_2}(x)=x(Bx+C) \cos 2x+x(Dx+E) \sin 2 x$ is a solution of the complimentary equation or auxiliary equation.
