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(1) $$y {y}^{\prime \prime}=x$$

Order $$=2$$
Degree $$=1$$

(2) $$\left(\frac{d^{4} y}{d x^{4}}\right)^{2}-2\left(\frac{d y}{d x}\right)^{3}=x-\sin x$$

Order $$=4$$
Degree $$=2$$

(3) $$x y^{(7)}-3 y^{\prime \prime}+\sin (x) y^{\prime}-4 y=10$$

Order $$= 7$$
Degree $$= 1$$

(4) $$\frac{1}{x} y^{(5)}-(\ln (x)+x) y^{\prime \prime}-(y-x) y^{\prime}=\frac{\sin x}{x}$$

Order $$=5$$
Degree $$=1$$

Determine the values of $$r$$ for which $$y=t^{r}, t>0$$ is a solution
of the differential equation $$t^{2} y^{\prime \prime}-4 t y^{\prime}+6 y=0$$

$$y=t^{r}$$

$$y^{\prime}=r t^{r-1}$$

$$y^{\prime \prime}=r(r-1) t^{r-2}$$

$$t^{2} y^{\prime \prime}-4 t y^{\prime}+6 y=0$$

$$t^{2}\left(r(r-1) t^{r-2}\right)-4 t\left(r t^{r-1}\right)+6 y=0$$

$$t^{2}\left(r^{2}-r\right) t^{r-2}-4 t\left(r t^{r-1}\right)+6 y=0$$

$$\left(r^{2}-r\right) t^{r}-4 r t^{r}+6 y=0$$

$$\left(r^{2}-r\right) t^{r}-4 r t^{r}+6 t^{r}=0$$

$$r^{2}-r-4 r+6=0$$

$$r^{2}-5 r+6=0$$

$$\Rightarrow(r-3)(r-2)=0$$

$$ {r=3} $$

$$ {r=2} $$

Assume that $$y_{1}(x) \neq 0$$ is a solution of $$y^{\prime}+p(x) y=q(x)$$ for $$x>0$$
Determine $$n$$ such that $$x y_{1}$$ is a solution of $$y^{\prime}+\left(p(x)-x^{n}\right) y=x q(x)$$

$$y_{1}$$ is a solution $$\longrightarrow y_{1}^\prime+p {y_{1}}=q$$

$$x y_{1}$$ is a solution $$\longrightarrow\left(x y_{1}\right)^\prime+\left(p-x^{n}\right)\left(x y_{1}\right)=x q$$

$$(1)\left(y_{1}\right)+\left(y_{1}^{\prime}\right)(x)+x p y_{1}-x^{n+1} y_{1}=x q$$

$$y_{1}^{\prime}(x)+x p y_{1}=x q$$

$$y_{1}^{\prime}+p y_{1}=q$$

$$y_{1}^{\prime}+p{y_{1}}=q$$

$$q=q$$

$$y_{1}-x^{n+1} y_{1}=0$$

$$\div y_{1}$$

$$\left[y_{1} \neq 0\right]$$

$$1-x^{n+1}=0$$

$$x^{n+1}=1 \quad \Rightarrow x^{n+1}=x^{0}$$

$$n+1=0 \Rightarrow n=-1$$

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