# Selected Solutions (IEL5-Quiz)

## Question 2

Let ϕ be the solution of the integral equation $\frac12\phi(x)-\int_0^1 e^{x-y}\phi(y)dy=x^2, (0\le x\le1$. Then

1. ϕ(0)=20exp(-1) – 8
2. ϕ(0)=20e-8
3. ϕ(0)=22 – 8e
4. ϕ(0)=22 – 8exp(-1)

### Solution 1

Let $c = \int_0^1 e^{-y}\phi(y)dy \implies \phi(x)=2x^2+2e^x c$. Substituting in the equation, $2x^2+2e^x c = 2x^2+ 2e^x\int_0^1 e^{-y}(2y^2+2e^y c)dy\implies c=\int_0^1e^{-y}(2y^2+ce^y)dy\implies c=2[\frac5e - 2]$

Hence, $\phi(x)=2x^2+4e^x [\frac5e - 2]$

Concluding, ϕ(0)=20exp(-1) – 8.

### Solution 2

For a Fredholm integral equation of 2nd Kind, using $\Gamma(s,t;\lambda)=\sum_p=0^\infty \frac{(-\lambda)^p}{p!}C_p(s,t)/ \sum_p=0^\infty \frac{(-\lambda)^p}{p!}c_p$, where

1. $c_0=1,C_0(s,t)=K(s,t)$
2. $c_p=\int_0^1 C_{p-1}(s,s)ds$
3. $C_p = c_pK(s,t)-p\int_0^1K(s,x)C_{p-1}(x,t)dx$

Here, $c_0=1,C_0(x,y)=e^{x-y},c_1=1,C_1(x,y)=0$. Hence $\Gamma(x,t;\lambda)=\frac{e^{x-y}}{1-2}=- e^{x-y}$

Hence the solution is $\phi(x)=2x^2+2\int_0^1\Gamma(x,y;\lambda)f(y)dy = 2x^2-4\int_0^1e^{x-y}y^2dy$ and you know how to finish the rest.

### Solution 3

Using iterated kernels, $\Gamma(s,t;\lambda)=\sum_{m=1}^\infty \lambda^{m-1}K_m(s,t)$

where $K_1(s,t) = K(s,t),K_m(s,t)=\int K(s,x)K_{m-1}(x,t)dx$

We will get $K_m(s,t) = e^{x-y}$, $\implies \Gamma(s,t;\lambda)=\sum_{m=1}^\infty \lambda^{m-1} e^{x-y}$ $= e^{x-y} \sum_{m=1}^\infty \lambda^{m-1}$ $= e^{x-y} \times (1-\lambda)^{-1}$

Here, $\lambda = 2\implies \Gamma(s,t;\lambda)= - e^{x-y}$ and you know where to take it from here.

### Observation

Use Leibniz Integral Rule (link to blog page quoting the result) – Wikipedia (In particular, a specific case called Fundamental Theorem of Calculus here, since limits are constant) to find derivative of the equation w.r to x. What do you observe? [Hint: you will get a differential equation, as $\phi'(x)-\phi(x)=4x-2x^2$ – try solving it!]

## Question 4

### Method

Solve the corresponding transposed homogenous integral equation, $\phi(x)=\frac2\pi\int_0^\pi \cos(x+t)\phi(t)dt$ and find corresponding solution $\psi(x)$ for $\lambda= \frac2\pi$.

Then, for each option, if $\int_0^\pi f(x)\psi(x)dx$. If it is zero, it has solutions, and if it is not zero, no solutions.

Here, solution, $\psi(x) =c_1\cos x +c_2 \sin x$ and hence options 2,3 and 4. ($\int_0^\pi (\cos x+\sin x)\cos(3x)dx=0$ and so on).