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$$t^{2} y^{\prime \prime}+2 t y^{\prime}-2 y=0, \quad t>0, \quad y_{1}(t)=t$$

$$y_{2}(t)=v(t) y_{1}(t) \quad y=v t$$

$$y^{\prime}=v^{\prime} t+v$$

$$y^{\prime \prime}=v^{\prime \prime} t+v^{\prime}+v^{\prime}$$

$$=v^{\prime \prime} t+2 v^{\prime}$$

$$t^{2}\left(v^{\prime \prime} t+2 v^{\prime}\right)+2 t\left(v^{\prime} t+v\right)-2 v t=0$$

$$t^{3} v^{\prime \prime}+2 t^{2} v^{\prime}+2 t^{2} v^{\prime}+2 t v-2 v t=0$$

$$t^{3} v^{\prime \prime}+4 t^{2} v^{\prime}=0$$

$$w=v^{\prime}, w^{\prime}=v^{\prime \prime} \quad t^{3} w^{\prime}+4 t^{2} w=0$$

$$t w^{\prime}+4 w=0$$

$$t w^{\prime}=-4 w$$

$$t \frac{d w}{d t}=-4 w$$

$$-\frac{1}{4} \frac{d w}{w}=\frac{d t}{t}$$

$$\frac{-1}{4} \ln |w|=\ln |t|+c$$

$$e^{-1 / 4 \ln |w|}=e^{\ln |t|+c}$$

$$\ln x^{n}=n \ln x$$

$$w^{-1 / 4}=e^{c} \cdot t$$


$$\left(v^{\prime}\right)^{-1 / 4}=c_{1} t$$

$$v^{\prime}=\left(c_{1} t\right)^{-4}$$


$$v^{\prime}=c_{2} t^{-4} \quad v=\frac{c_{2} t^{-3}}{-3}+c_{3}$$

$$c_{2} /-3=c_{4}$$

$$v=c_{4} t^{-3}+c_{3}$$

$$y_{2}=c_{4} t^{-2}+c_{3} t$$

Reduction of order

$$4 y^{\prime \prime}+17 y^{\prime}+4 y=0$$

$$4 r^{2}+17 r+4=0$$

$$r_{1}=-1 / 4, \quad r_{2}=-1$$

$$y=c_{1} e^{{-1/4} t}+c_{2} e^{-4 t}$$

$$y^{\prime \prime}-2 y^{\prime}+10 y=0$$

$$r^{2}-2 r+10=0$$

$$r=1 \pm 3 i \quad \lambda=1, \mu=3$$

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

$$9 y^{\prime \prime}+6 y^{\prime}+y=0$$

$$9 r^{2}+6 r+1=0$$

$$r=-1 / 3=r_{1}=r_{2}$$

$$y=c_{1} e^{-t / 3}+c_{2} t e^{-t / 3}$$

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