Cálculo C

Lista 10

Exercício 1   Achar a solução geral de
$$\begin{align*}a)\;&y'''-2y''-3y'=0 & R:y=C_1+C_2e^{-x}+C_3e^{3x}\\ b)\;&y'''+6y... ...xt {\ sen }}x\\ h)\;&y^{(X)}=0 & R:y=C_1+C_2x+\cdots C_{10}x^9\\ \end{align*}$$

Exercício 2   Resolver os problemas de valores iniciais:

1.

$$\begin{align*}&y''-4y'+3y=0& \\ &y(0)=6,\;\; y'(0)=10 & R:y=4e^x+2e^{3x} \end{align*}$$

2.

$$\begin{align*}&y'''-3y''+3y'-y=0& \\ &y(0)=1,\;\; y'(0)=2,\;\;y''(0)=3 & R:y=e^x(1+x) \end{align*}$$

3.

$$\begin{align*}&y''-2y''+3y=0& \\ &y(0)=1,\;\; y'(0)=3 & R:y=e^x(\cos \sqrt{2}x+{\text {\ sen }}\sqrt{2}x) \end{align*}$$

4.

$$\begin{align*}&y''-2y''+2y=0& \\ &y(0)=0,\;\; y'(0)=1 & R:y=e^x{\text {\ sen }}x \end{align*}$$

5.

$$\begin{align*}&y'''-y=2x& \\ &y(0)=0,\;\; y'(0)=0\;\;y''(0)=2 & R:y=-\frac{4}{\sqrt{3}}e^{-x/2}{\text {\ sen }} x\frac{\sqrt{3}}{2}+2x \end{align*}$$

6.

$$\begin{align*}&y''-6y'+9y=x^2-x+3& \\ &y(0)=2,\;\; y'(0)=8 & R:y=\frac{1}{27}\left[(31+118x)e^{3x}+13+7x+3x^2\right] \end{align*}$$

Exercício 3   Achar uma solução particular de:
$$\begin{align*}a)\;&y'''+y''=3 & R:y_p=x^2\\ b)\;&y^{iv}+2y'''+y''=xe^{-x} &R:y_p=x^2e^{-x}\\ \end{align*}$$

Exercício 4   Achar a solução geral de:

$$\begin{aligned} a)\;&4y''+8y'=x{\text {\ sen }}x\\ b)\;&x^2y... ... {\ sen }}x-x^2-3x-1\\ R:y=C_1+C_2x^2+C_3x^4\\ \end{gathered}$$

Exercício 5   Achar a solução de x²y''+2xy'-6y=0cujo gráfico passa pelo ponto (1,1).

R: $y=\frac{3}{5}x^2+\frac{2}{5}x^{-3}.$