Differential EquationComplex Roots Problems

$$y^{\prime \prime}-2 y^{\prime}+6 y=0$$

$$r^{2}-2 r+6=0$$

$$r=1 \pm i \sqrt{5} \quad \lambda=1 \quad \mu=\sqrt{5}$$

$$y=c_{1} e^{\lambda t} \cos \mu t+c_{2} e^{\lambda t} \sin \mu t$$

$$y=c_{1} e^{t} \cos \sqrt{5} t+c_{2} e^{t} \sin \sqrt{5} t$$

$$y^{\prime \prime}+4 y+6.25 y=0$$

$$r^{2}+4 r+6.25=0$$

$$r=-2 \pm \frac{3}{2} i$$

$$\lambda=-2 \quad, \mu=3 / 2$$

$$y=c_{1} e^{-2 t} \cos \frac{3 t}{2}+c_{2} e^{-2 t} \sin \frac{3 t}{2}$$

$$y^{\prime \prime}-2 y^{\prime}+5 y=0 \quad, y\left(\frac{\pi}{2}\right)=0, \quad y^{\prime}\left(\frac{\pi}{2}\right)=2$$

$$r^{2}-2 r+5=0 \quad r=1 \pm 2 i$$

$$y=c_{1} e^{t} \cos 2 t+c_{2} e^{t} \sin 2 t$$

$$0=c_{1} e^{\pi / 2} \cos \pi+c_{2} e^{\pi / 2} \sin \pi$$

$$0=c_{1} e^{\pi/{2}}(-1)+0 \Rightarrow c_{1}=0$$

$$y=c_{2} e^{t} \sin 2 t$$

$$y^{\prime}=c_{2}\left[e^{t} \sin 2 t+(2 \cos 2 t)\left(e^{t}\right)\right]$$

$$2=c_{2}\left[e^{\pi / 2} \sin \pi+(2 \cos \pi)\left(e^{\pi / 2}\right)\right]$$

$$2=c_{2}\left(-2 e^{\pi / 2}\right)$$

$$1=c_{2}\left(-e^{\pi / 2}\right) \Rightarrow c_{2}=\frac{-1}{e^{\pi / 2}}$$

$$c_{2}=-e^{-\pi / 2}$$

$$y=-e^{-\pi / 2} e^{t} \sin 2 t$$

$$y=-e^{t-\pi / 2} \sin 2 t$$