Lista VI

Cálculo D (E)

Exercício 1   Exprima a solução geral de cada uma das equações seguintes como uma série de potências com centro em x=0.

a) $\ddot{y}-3xy=0$                 $y=a_0\, y_0+a_1\,y_1$

$$y=1+ \sum\limits_{k=1}^\infty\frac{1}{[2\cdot 5\cdot 8\cdots (3k-1)]k!}x^{3k}$$

$$y_2= \sum\limits_{k=0}^\infty\frac{1}{[1\cdot 4\cdot 7\cdot \cdots (3k+1) k!}]x^{3k+1}$$

b) $(x^2+1)\ddot{y}-6y=0$

$$y=a_1(x+x^3)+3a_0\left( 1+ \sum\limits_{k=1}^\infty\frac{(-1)^k}{(2k-3)(2k-1)}x^{2k} \right)$$

c) $(2x^2+1)\ddot{y}+6xy-18y=0$

$$y=a_1(x+\frac{8}{3}x^3)+ a_0\left[ 1+ 9x^2+\frac{15}{2}x^4+3\... ...ty (-1)^k\frac{(2k-5)!(2k+1)}{2^{k-3}[k!(k-3)!]}x^{2k} \right]$$

d) $\dddot{y}-3x\dot{y}-y=0$         $y=a_0\,y_0+a_1\,y_1+a_2\,y_2$

$$y_0=1+\sum\limits_{k=1}^\infty \frac{1\cdot 10\cdot 19\cdots (9k-8)}{(3k)!}x^{3k}$$

$$y_1=x+\sum\limits_{k=1}^\infty \frac{4\cdot 13\cdot 22\cdots (9k-5)}{(3k+1)!}x^{3k+1}$$

$$y_2=x^2+2\sum\limits_{k=1}^\infty \frac{7\cdot 16\cdot 25\cdots (9k-2)}{(3k+2)!}x^{3k+2}$$

Exercício 2   Resolva os problemas de valor inicial:

a)

$$\ddot{y}+x\dot{y}-2y=e^x$$

$$y(0)=0\quad \dot{y}(0)=0$$

b)

$$3\ddot{y}-\dot{y}+(x+1)y=1$$

$$y(0)=0\quad \dot{y}(0)=0$$

Solução:

$$y=\frac{1}{6}x^2+\frac{1}{54}x^3-\frac{1}{324}x^4-\frac{4}{1215}x^5+\cdots$$

c)

$$(1-x^2)\ddot{y}-2x \dot{y}+\lambda(\lambda+1)y=0$$

$$y(0)=a_0\quad \dot{y}(0)=a_1$$

Solução: $y=a_o\,y_0+a_1\,y_1$ onde

$$y_0=1-\frac{(\lambda+1)}{2!}x^2+ \frac{(\lambda+3)(\lambda+2)(\lambda-2)}{4!}x^4-\cdots$$

$$y_1=x-\frac{(\lambda+2)(\lambda-1)}{3!}x^3+ \frac{(\lambda+4)(\lambda+2)(\lambda-1)(\lambda-1)(\lambda-3)}{5!}x^5-\cdots$$