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).

Part A: Classical (Exact) Solution (20 marks)

Step 1: Identify the differential equation.
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 $\beta$).

Step 2: Multiply by $\dot{x}$ to get energy conservation.
$\dot{x} \ddot{x} + \alpha x \dot{x} + \beta x^3 \dot{x} = 0.$
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).

Step 3: Separate variables.
From the energy equation,
$\dot{x} = \pm \sqrt{2 \left( E - \frac{\alpha}{2} x^2 - \frac{\beta}{4} x^4 \right)}.$
Thus,
$dt = \frac{dx}{\pm \sqrt{2 \left( E - \frac{\alpha}{2} x^2 - \frac{\beta}{4} x^4 \right)}}.$

Step 4: Integrate to find the period or solution form.
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 ($E = \frac{\alpha}{2} A^2 + \frac{\beta}{4} A^4$).

Step 5: Recognize elliptic integral form.
This is an elliptic integral of the first kind. The exact solution is
$x(t) = x_0 \, cn \left( \Omega (t - t_0) \mid k \right),$
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.)

Part B: Any Two Classical Methods (10 marks)

Method 1: Euler's Method
Step 1: Rewrite as system: Let $v = \dot{x}$, then
$\dot{x} = v, \quad \dot{v} = -\alpha x - \beta x^3.$

Step 2: Euler update:
$x_{n+1} = x_n + h v_n,$
$v_{n+1} = v_n + h (-\alpha x_n - \beta x_n^3),$
where $h$ is step size.
Simple, explicit, first-order accurate, but unstable for stiff problems.

Method 2: Classical Fourth-Order Runge-Kutta (RK4)
Step 1: Same system $\mathbf{y}' = f(t, y)$, $\mathbf{y} = [x, v]^T$.

Step 2: RK4 increments:
$k_1 = h f(t_n, y_n),$
$k_2 = h f(t_n + \frac{h}{2}, y_n + \frac{k_1}{2}),$
$k_3 = h f(t_n + \frac{h}{2}, y_n + \frac{k_2}{2}),$
$k_4 = h f(t_n + h, y_n + k_3).$
Then
$\mathbf{y}_{n+1} = y_n + \frac{1}{6} (k_1 + 2k_2 + 2k_3 + k_4).$
Fourth-order accurate, good for nonlinear ODEs like Duffing.

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