Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$

Question

Integral equation

$$y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$$ has:

a unique solution for $\lambda \neq \frac{4}{\pi +2}$;

a unique solution for $\lambda \neq \frac{4}{\pi -2}$;

no solution for $\lambda \neq \frac{4}{\pi +2}$, but the corresponding homogeneous equation has a non-trivial solution; or

no solution for $\lambda \neq \frac{4}{\pi -2}$, but the corresponding homogeneous equation has a non-trivial solution.

I am stuck on this problem. Can anyone help me please?

If it is a question of some Indian Entrance Examination please add the source (Exam. name, year)in the title of this question. It will be helpful to other students using this site. — Supriyo, Jan 21 '14 at 11:06

Nameless · Answer 1 · 2012-12-18T16:33:29.790

3

Hint: Use $$\cos(x-t)=\cos x\cos t+\sin x\sin t$$ and derive twice. After that you get: $$y^{\prime\prime}(x)=-(\lambda\int_{0}^2\cos(x-t)y(t)dt)$$ and so $$y^{\prime\prime}+y-1=0$$ The corresponding homogenous equation is $$y^{\prime\prime}+y=0$$ The solution is $y=a\cos x+b\sin x+1$. We just have to subsititute it back in the original equation. Then, $$a\cos x+b\sin x+1=1+\lambda \int_{0}^2\cos(x-t)(a\cos t+b\sin t+1)dt$$ and so $$a\cos x+b\sin x=\lambda \cos x\int_{0}^2\cos t(a\cos t+b\sin t+1)dt+\\\lambda \sin x\int_{0}^2\sin t(a\cos t+b\sin t+1)dt$$ Therefore, we have the system of equations: \begin{gather}a=\lambda \int_{0}^2\cos t(a\cos t+b\sin t+1)dt\\ b=\lambda \int_{0}^2\sin t(a\cos t+b\sin t+1)dt\end{gather}

edited Dec 18 '12 at 16:33

answered Dec 18 '12 at 14:55

Nameless

13,456

$a\doteq \lambda \sin t \int_0^2 (a\cos t \sin t + b\sin^2 t + \sin t) dt $.it getting hard – pankaj Dec 18 '12 at 16:36
its getting hard ,i have to then solve for homogeneous also,is there any shortcut........ – pankaj Dec 18 '12 at 16:48
@pankaj I don't know any shortcut you can take other than computing the integrals, that are by the way easier than you think. Look at Mhenni's answer for more info – Nameless Dec 18 '12 at 17:02
sir tell me what I do after finding a,b – pankaj Dec 18 '12 at 17:06
non trivial means what sir........... – pankaj Dec 18 '12 at 17:07
@pankaj You will find $a,b$ with respect to $\lambda$. Now a trivial solution to the homogenous is $y=0$. The problems asks for other solutions. – Nameless Dec 18 '12 at 17:08
OKK its somewhat lengthy.......... – pankaj Dec 18 '12 at 17:10

score 2 · Answer 2 · edited Apr 13 '17 at 12:20

2

Related technique: (I), (II). The integral equation you have is a "Fredholm equation of the 2nd kind with seperable kernel". There are standard techniques to solve this type of equations. See here, page 20 for the method and a worked example how to find such $\lambda$.

edited Apr 13 '17 at 12:20

Community

1

answered Dec 18 '12 at 16:01

Mhenni Benghorbal

47,431

its hard ,i have to solve for homogeneous and non homogeneous also,is there any shortcut..... – pankaj Dec 18 '12 at 16:50
@pankaj: Where did this problem come from? – Mhenni Benghorbal Dec 18 '12 at 17:13
@pankaj: From what course has this problem come from? – Mhenni Benghorbal Dec 19 '12 at 14:55
This is indeed a Fredholm equation of the 2nd kind with seperable kernel... but the fact that is irrelevant. Please try not to send the OP on deadends. – Did Dec 25 '12 at 19:08

Integral equation solution: $y(x) = 1 + \lambda\int\limits_0^2\cos(x-t) y(t) \mathrm{d}t$

2 Answers2

Linked