10.9.2 习题

尽可能使用 MATLAB 检验你的答案。

55 求下列微分方程的通解：

(a) $2\frac{d^2x}{dt^2} - 5\frac{dx}{dt} + 3x = 0$

(b) $\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 5x = 0$

(c) $\frac{d^2x}{dt^2} + 3\frac{dx}{dt} - 4x = 0$

(d) $\frac{d^2x}{dt^2} - 4\frac{dx}{dt} + 13x = 0$

56 求解下列初值问题：

(a) $5\frac{d^2x}{dt^2} - 3\frac{dx}{dt} - 2x = 0, x(0) = -1, \frac{dx}{dt}(0) = 1$

(b) $\frac{d^2x}{dt^2} - 6\frac{dx}{dt} + 10x = 0, x(0) = 2, \frac{dx}{dt}(0) = 0$

(c) $\frac{d^2x}{dt^2} - 4\frac{dx}{dt} + 3x = 0, x(0) = 0, \frac{dx}{dt}(0) = 1$

57 求下列微分方程的通解：

(a) $4\frac{d^2x}{dt^2} - 2\frac{dx}{dt} + 7x = 0$

(b) $\frac{d^2x}{dt^2} + 6\frac{dx}{dt} - 4x = 0$

(c) $3\frac{d^2x}{dt^2} + 3\frac{dx}{dt} + 3x = 0$

(d) $\frac{d^2x}{dt^2} - 8\frac{dx}{dt} + 16x = 0$

(e) $9\frac{d^3x}{dt^3} - 9\frac{d^2x}{dt^2} - 4\frac{dx}{dt} + 4x = 0$

(f) $\frac{d^3x}{dt^3} - \frac{d^2x}{dt^2} + 7\frac{dx}{dt} + 9x = 0$

(g) $\frac{d^3x}{dt^3} - 2\frac{d^2x}{dt^2} + 3\frac{dx}{dt} = 0$

58 证明微分方程的特征方程为

$$\frac{d^4x}{dt^4} - 4\frac{d^3x}{dt^3} + 11\frac{d^2x}{dt^2} - 14\frac{dx}{dt} + 10x = 0$$

并且

$$(m^2 - 2m + 2)(m^2 - 2m + 5) = 0$$

从而求出该方程的通解。

59 求解下列初值问题：

(a) $2\frac{d^2x}{dt^2} - 2\frac{dx}{dt} + 3x = 0, x(0) = 1, \frac{dx}{dt}(0) = 0$

(b) $\frac{d^2x}{dt^2} - 4\frac{dx}{dt} + 4x = 0, x(1) = 0, \frac{dx}{dt}(1) = 2$

(c) $\frac{d^2x}{dt^2} + 5\frac{dx}{dt} + 8x = 0, x(0) = 1, \frac{dx}{dt}(0) = -2$

(d) $9\frac{d^2x}{dt^2} + 6\frac{dx}{dt} + x = 0, x(-3) = 2, \frac{dx}{dt}(-3) = \frac{1}{2}$

(e) $\frac{d^3x}{dt^3} - 6\frac{d^2x}{dt^2} + 11\frac{dx}{dt} - 6x = 0,$

$$x(0) = 1, \frac{dx}{dt}(0) = 0, \frac{d^2x}{dt^2}(0) = 1$$

(f) $\frac{d^3x}{dt^3} + 6\frac{d^2x}{dt^2} + 12\frac{dx}{dt} + 8x = 0,$

$$x(1) = 1, \frac{dx}{dt}(1) = 1, \frac{d^2x}{dt^2}(1) = 0$$

60 证明微分方程的特征方程为

$$\frac{d^4x}{dt^4} - 2\frac{d^3x}{dt^3} + 3\frac{d^2x}{dt^2} - 2\frac{dx}{dt} + x = 0$$

并且

$$(m^2 - m + 1)^2 = 0$$

从而求出该方程的通解。

61 证明微分方程的特征方程为

$$\frac{d^4x}{dt^4} - \frac{d^3x}{dt^3} - 9\frac{d^2x}{dt^2} - 11\frac{dx}{dt} - 4x = 0$$

是

$$(m^3 + 3m^2 + 3m + 1)(m - 4) = 0$$