A) Discuss the i) theory of classical condition (10 marks) and ii) the theory of operant condition (10 marks). b) Discuss how the two theories can be applied in a life of learner (20 marks).

The Duffing equation (undamped, unforced case) is

$$\ddot{x} + \alpha x + \beta x^3 = 0,$$

where typically $\alpha > 0$ (hardening or softening spring depending on $\beta$).

$$\dot{x} \ddot{x} + \alpha x \dot{x} + \beta x^3 \dot{x} = 0.$$

Integrate with respect to $t$:

$$\frac{1}{2} \dot{x}^2 + \frac{\alpha}{2} x^2 + \frac{\beta}{4} x^4 = E,$$

where $E$ is a constant (total energy).

From the energy equation,

$$\dot{x} = \pm \sqrt{2 \left( E - \frac{\alpha}{2} x^2 - \frac{\beta}{4} x^4 \right)}.$$

Thus,

$$dt = \frac{dx}{\pm \sqrt{2 \left( E - \frac{\alpha}{2} x^2 - \frac{\beta}{4} x^4 \right)}}.$$

The solution is periodic. The quarter-period $T/4$ is

$$\frac{T}{4} = \int_0^{A} \frac{dx}{\sqrt{2 \left( E - \frac{\alpha}{2} x^2 - \frac{\beta}{4} x^4 \right)}},$$

where $A$ is the amplitude ($E = \frac{\alpha}{2} A^2 + \frac{\beta}{4} A^4$).

This is an elliptic integral of the first kind. The exact solution is

$$x(t) = x_0 \, \mathrm{cn} \left( \Omega (t - t_0) \mid k \right),$$

where $\mathrm{cn}$ is the Jacobi elliptic cosine function, and parameters $\Omega$, $k$ depend on $\alpha$, $\beta$, $x_0$ (amplitude):

$$k = \sqrt{\frac{\beta x_0^2}{2(\alpha + \beta x_0^2)}}, \quad \Omega = \sqrt{\alpha + \beta x_0^2} \sqrt{1 + k^2}.$$

(This is the classical exact solution using elliptic functions.)

Let $v = \dot{x}$, then

$$\dot{x} = v, \quad \dot{v} = -\alpha x - \beta x^3.$$

$$x_{n+1} = x_n + h v_n,$$ $$v_{n+1} = v_n + h (-\alpha x_n - \beta x_n^3),$$

where $h$ is step size.
Simple, explicit, first-order accurate, but unstable for stiff problems.

Method 2: Classical Fourth-Order Runge-Kutta (RK4)

$$k_1 = h \mathbf{f}(t_n, \mathbf{y}_n),$$ $$k_2 = h \mathbf{f}(t_n + \frac{h}{2}, \mathbf{y}_n + \frac{k_1}{2}),$$ $$k_3 = h \mathbf{f}(t_n + \frac{h}{2}, \mathbf{y}_n + \frac{k_2}{2}),$$ $$k_4 = h \mathbf{f}(t_n + h, \mathbf{y}_n + k_3).$$

Then

$$\mathbf{y}_{n+1} = \mathbf{y}_n + \frac{1}{6} (k_1 + 2k_2 + 2k_3 + k_4).$$

Fourth-order accurate, good for nonlinear ODEs like Duffing.

x(t) = x_0 \cn(\Omega(t - t_0) \mid k)