4.48 Proof (iii) Complex conjugate roots

(iii) The roots are $\alpha\pm i\beta$, so auxiliary equation is

$$(s-\alpha-i\beta)(s-\alpha+i\beta)=0,$$

$$s^{2}-2\alpha s+\alpha^{2}+\beta^{2}=0.$$

Consider

$$w(x)=e^{-\alpha x}y(x)$$

$$w^{\prime}(x)=e^{-\alpha x}y^{\prime}(x)-\alpha e^{-\alpha x}y(x)$$

$$w^{\prime\prime}(x)=e^{-\alpha x}y^{\prime\prime}(x)-2\alpha e^{-\alpha x}y^{\prime}(x)+\alpha^{2}e^{-\alpha x}y(x)$$

where

$$y^{\prime\prime}(x)-2\alpha y^{\prime}(x)+(\alpha^{2}+\beta^{2})y(x)=0,$$

so

$$w^{\prime\prime}(x)+\beta^{2}w(x)=0.$$

Then by SHM solution $w(x)=A\cos\beta x+B\sin\beta x$, so

$$y(x)=e^{\alpha x}(A\cos\beta x+B\sin\beta x).$$