(i) State the Fredholm alternative for Fredholm integral equations of the second kind.

Show that the integral equation

$$\phi(x)-\lambda \int_{0}^{1}(x+t) \phi(t) d t=f(x), \quad 0 \leqslant x \leqslant 1$$

where $f$ is a continuous function, has a unique solution for $\phi$ if $\lambda \neq-6 \pm 4 \sqrt{3}$. Derive this solution.

(ii) Describe the WKB method for finding approximate solutions $f(x)$ of the equation

$$\frac{d^{2} f(x)}{d x^{2}}+q(\epsilon x) f(x)=0$$

where $q$ is an arbitrary non-zero, differentiable function and $\epsilon$ is a small parameter. Obtain these solutions in terms of an exponential with slowly varying exponent and slowly varying amplitude.

Hence, by means of a suitable change of independent variable, find approximate solutions $w(t)$ of the equation

$$\frac{d^{2} w}{d t^{2}}+\lambda^{2} t w=0$$

in $t>0$, where $\lambda$ is a large parameter.