Determining the number of solutions of the non homogeneous integral equation.

Question

This is a question from a competitive exam. We are given the integral equation: $$ \phi (x) = \cos(7x) + \lambda \int_{0}^{\pi} \left[ \cos(x)\cos(t) - 2\sin(x)\sin(t) \right]\phi(t) dt $$ and are asked the number of solutions of the equation depending on the value of lambda. The options are:

1. Solution exists for every value of $\lambda$.

2. There is some $\lambda$ for which solution does not exist.

3. There is some $\lambda$ for which more than one but finitely many solutions exist.

4.There is $\lambda$ such that infintely many solution exists.

Now after refotmulation it in the form of matrix I have obtained: $\left| \begin{matrix} 1- \frac{ \lambda \pi}{2} & 0\\ 0 & 1 + \lambda \pi \end{matrix} \right| =0 $ This gives me the values of $\lambda$. But I can not conclude the number of solutions from here. Any help is highly appreciated.

Answer 1

$$ \phi (x) = \cos(7x) + \lambda \int_{0}^{\pi} \left[ \cos(x)\cos(t) - 2\sin(x)\sin(t) \right]\phi(t) dt $$ A different approach : $$ \phi (x) = \cos(7x) + \lambda \cos(x)\int_{0}^{\pi} \cos(t)\phi(t)dt -2\lambda \sin(x)\int_{0}^{\pi} \sin(t)\phi(t) dt $$ The integrals are not function of $x$. They are constant.

Let$\quad \int_{0}^{\pi} \cos(t)\phi(t)dt=A\quad$ and $\quad \int_{0}^{\pi} \sin(t)\phi(t) dt=B$

$$ \phi (x) = \cos(7x) + \lambda A \cos(x)-2\lambda B \sin(x) $$

$$A=\quad \int_{0}^{\pi} \cos(t)\left(\cos(7t) + \lambda A \cos(t)-2\lambda B \sin(t) \right)dt=\frac{\pi}{2}\lambda A \quad\implies\quad (1-\frac{\pi}{2}\lambda)A=0$$

$$B=\quad \int_{0}^{\pi} \sin(t)\left(\cos(7t) + \lambda A \cos(t)-2\lambda B \sin(t) \right)dt=-\pi\lambda B \quad\implies\quad (1+\pi\lambda)B=0$$

In general case (any $\lambda$ ) if $\lambda\neq \frac{2}{\pi}$ and if $\lambda\neq\frac{-1}{\pi}$ this implies $A=0$ and $B=0$

Thus the solution is : $$\boxed{\Phi(x)=\cos(7x)}$$

The two particular cases $\lambda=\frac{2}{\pi}$ or $\lambda=\frac{-1}{\pi}$ have to be studied separately. Each one will lead to an infinity of solutions with arbitrary $A$ or $B$ respectively.

$ \phi (x) = \cos(7x) + \frac{2}{\pi} A \cos(x)\quad $ if $\lambda= \frac{2}{\pi}$.

$ \phi (x) = \cos(7x) + \frac{2}{\pi} B \sin(x)\quad $ if $\lambda= \frac{-1}{\pi}$.