This is actually a problem in algebra as shall be seen. I need to find the general solution for the following differential equation:
$$y''''+8y''+16y=0$$
The characteristic equation for this is:
$$\lambda^4+8\lambda^2+16=0$$
Factoring out gives us:
$$(\lambda^2+4)^2=(\lambda^2+4)(\lambda^2+4)=0$$
This generates a set of double complex conjugate roots $\lambda_{1,2}=\pm i2$ and $\lambda_{3,4}=\pm i2$
The general solution I get is:
$$y=A\cos(2x)+B\sin(2x)+xC\cos(2x)+xD\sin(2x)$$
Is this correct? If not please explain in detail where I went wrong. Thank you so much.
Best Answer
This solution is correct. Good job!