解法 由于这是一个三阶方程，我们需要引入两个新变量：

$$y = \frac{dx}{dt} \quad \text{and} \quad z = \frac{dy}{dt} = \frac{d^2x}{dt^2}$$

于是该方程被转化为由三个一阶微分方程组成的方程组

$$\frac{dx}{dt} = y \qquad x(0) = 1$$

$$\frac{dy}{dt} = z \qquad y(0) = 0.5$$

$$\frac{dz}{dt} = -xtz - ty + t^2x \quad z(0) = -0.2$$

将该格式应用于这组微分方程

$$\frac{dx}{dt} = f_1(t, x, y, z)$$

$$\frac{dy}{dt} = f_2(t, x, y, z)$$

$$\frac{dz}{dt} = f_3(t, x, y, z)$$

欧拉格式具有如下形式

$$X_{n+1} = X_n + hf_1(t_n, X_n, Y_n, Z_n)$$

$$Y_{n+1} = Y_n + hf_2(t_n, X_n, Y_n, Z_n)$$

$$Z_{n+1} = Z_n + hf_3(t_n, X_n, Y_n, Z_n)$$

因此，在本例中我们有

$$X_0 = x_0 = 1$$

$$Y_0 = y_0 = 0.5$$

$$Z_0 = z_0 = -0.2$$

$$f_1(t_0, X_0, Y_0, Z_0) = Y_0 = 0.5000$$

$$f_2(t_0, X_0, Y_0, Z_0) = Z_0 = -0.2000$$

$$f_3(t_0, X_0, Y_0, Z_0) = -X_0t_0Z_0 - t_0Y_0 + t_0^2X_0$$

$$= -1.0000 \times 0 \times (-0.2000) - 0 \times 0.5000 + 0^2 \times 1.0000$$

= 0.0000

$$X_1 = 1.0000 + 0.05 \times 0.5000 = 1.0250$$

$$Y_1 = 0.5000 + 0.05 \times (-0.2000) = 0.4900$$

$$Z_1 = -0.2000 + 0.05 \times 0.0000 = -0.2000$$

$$f_1(t_1, X_1, Y_1, Z_1) = Y_1 = 0.4900$$