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$$f''+\lambda f = 0, \quad f=f(y), \quad a \leq y \leq b, \quad f(a)=f(b)=0 $$

In this case, I tried

$$f(y)=sin(k_ny), \quad \lambda=k_n^2 $$ $$f(a)=0 \Longrightarrow k_na = n\pi, \quad n=0,1,2,...$$ $$ k_n=\frac{n\pi}{a}, \quad n=0,1,2,... $$

The problem is that I don't know how to apply the second boundary condition here, so I know something in my approach is wrong. The answer is stated below. I can see why this solution works, but I just don't know how to derive it myself. Also I'm wondering if there is a solution that works from $n=0$.

$$ f(y)=sin(k_n(y-a)), \quad k_n=\frac{n\pi}{b-a}, \quad n=1,2,... $$

EDIT: I tried setting $$ f(y) = Acos(k_ny) + Bsin(k_ny) $$ $$ f(a) = Acos(k_na) + Bsin(k_na) = 0 $$ $$ f(b) = Acos(k_nb) + Bsin(k_nb) = 0 $$

But I'm not sure where to go from here. $f(a)$ can be zero if $A=-B, k_na = \pi/4 + n\pi$, but it can also be zero if $A=0, k_na=n\pi$ or $B=0, k_na=\pi/2+n\pi$. Do I have to check the second condition for all these 3 solutions, or is there any way to know which one I should go with?

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    $\begingroup$ The function you should try is $f(y) = C_1 \cos (k_n y) + C_2 \sin (k_n y)$ $\endgroup$ – JoseSquare Feb 25 at 20:12
  • $\begingroup$ $f_0 (y)=0$ obviously satisfies both the equation and boundary conditions. $\endgroup$ – user Feb 25 at 20:54
  • $\begingroup$ @user but in the answer, $n=0$ isn't defined. so my second question was if there's a way to rewrite the answer so it's defined for $n=0$ aswell. this is not as important as understanding how they reach the answer though, it's 2nd prio $\endgroup$ – armara Feb 26 at 7:47
  • $\begingroup$ @JoseSquare hey thx for the tip, I updated the answer with an EDIT $\endgroup$ – armara Feb 26 at 7:47
  • $\begingroup$ Contrary to what you think, there are no distinct $k_n$ other than $\sqrt\lambda$. $\endgroup$ – Yves Daoust Feb 26 at 8:10
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Assuming $\lambda>0$, the general solution is

$$f(y)=c_c\cos(\sqrt\lambda y)+c_s\sin(\sqrt\lambda y).$$

Plugging the boundary conditions,

$$0=c_c\cos(\sqrt\lambda a)+c_s\sin(\sqrt\lambda a), \\0=c_c\cos(\sqrt\lambda b)+c_s\sin(\sqrt\lambda b).$$

The determinant of this homogeneous linear system is

$$\sin(\sqrt\lambda(b-a)).$$

So

  • if $\sqrt\lambda(b-a)$ is not a multiple of $\pi$, only the trivial solution $f=0$ is possible;

  • if $\sqrt\lambda(b-a)$ is a multiple of $\pi$, the solutions are

$$c\sin(\sqrt\lambda(x-a))$$ where $c$ is free.

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  • $\begingroup$ this was an approach i hadn't thought about, need to read a bit more about how determinants can be used to solve system of equations. thanks for bringing this to my attention, i gave you +1! $\endgroup$ – armara Feb 26 at 8:16
  • $\begingroup$ @armara: you'd better accept this answer. It does give the complete solution. [I never ask such things, but here it's too blatant.] You seem to be stuck on the idea that there are several modes, which is false. $\endgroup$ – Yves Daoust Feb 26 at 8:19
  • $\begingroup$ sure I can do that, you obviously know this better than me. however, i'm personally going to use the other answer for now since I don't quite understand this one. i'll be studying this answer tho, seems like it's good :) if you have any pointers on where i should read to understand these concepts better, i would gladly go there $\endgroup$ – armara Feb 26 at 8:26
  • $\begingroup$ @armara: I can't believe that you don't know the resolution of a $2\times2$ system. You are just disconcerted by the trigonometric functions. $\endgroup$ – Yves Daoust Feb 26 at 8:29
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Hint: Solve the related problem

$$ f''(x) + \lambda f(x) = 0, \quad 0 \le x \le 1, \quad f(0)=f(1)=0 $$

Then make the coordinate transform $y = a + (b-a)x $

Edit: You can think about the transform organically. The general solution is

$$ f(y) = A\cos(\sqrt{\lambda}y) + B\sin(\sqrt{\lambda}y) \tag{1} $$

This would be very convenient if one of the B.C.s was at $x=0$ since you have $f(0) = A$ and $f'(0)=\sqrt{\lambda}B$. Since this is not the case, you can try to see what would happen if the solution was shifted to $x=a$ instead

$$ f(y) = \widetilde{A}\cos(\sqrt{\lambda}(y-a)) + \widetilde{B}\sin(\sqrt{\lambda}(y-a)) \tag{2} $$

This is also a solution of the ODE since the operation $f''$ is shift-invariant, and the shifted sinusoids are still a linear combination of the original $\sin$ and $\cos$ functions.

Using $(2)$ as the solution form gives $\widetilde{A} = 0$ and $\sqrt{\lambda} = \dfrac{n\pi}{b-a}$.

Edit 2: I will expand on Yves' answer further to show how you can arrive at the same answer

Using $(1)$ as the solution form, we need to solve

\begin{align} A\cos(\sqrt{\lambda}a) + B\sin(\sqrt{\lambda}a) &= 0 \\ A\cos(\sqrt{\lambda}b) + B\sin(\sqrt{\lambda}b) &= 0 \end{align}

We'll get a non-trivial solution if the determinant of this system is $0$. Another way to understand this without using the determinant is to solve the system normally. Find $B$ in terms of $A$ in the second equation and plug it into the first, you get

$$ A\cos(\sqrt{\lambda}a) - A\frac{\cos(\sqrt{\lambda}b)}{\sin(\sqrt{\lambda}b)}\sin(\sqrt{\lambda}a) = 0 $$

$$ \implies A\big[\cos(\sqrt{\lambda}a)\sin(\sqrt{\lambda}b) - \cos(\sqrt{\lambda}b)\sin(\sqrt{\lambda}a)\big] = A\sin(\sqrt{\lambda}(b-a)) = 0 $$

We can't have $A=0$, so we require

$$ \sin\sqrt{\lambda}(b-a) = 0 \implies \sqrt{\lambda}(b-a) = n\pi $$

where $n=1,2,3,\dots$, $n$ can't be $0$ here since it would also give a trivial solution (you should've excluded the cases $\lambda=0$ and $\lambda < 0$ earlier on)

Now, $A$ and $B$ form a family of solutions satisfying

$$ \frac{A}{B} = -\frac{\sin(\sqrt{\lambda}a)}{\cos(\sqrt{\lambda}a)} = -\frac{\sin(\sqrt{\lambda}b)}{\cos(\sqrt{\lambda}b)} $$

we can arbitrarily set $A = -c\sin(\sqrt{\lambda}a)$, $B=c\cos(\sqrt{\lambda}a)$ (which is fine since we know the final solution still has one free constant). Thus

$$ f(y) = -c\sin(\sqrt{\lambda}a)\cos(\sqrt{\lambda}y) + c\cos(\sqrt{\lambda}a)\sin(\sqrt{\lambda}y) = c\sin(\sqrt{\lambda}(y-a)) $$

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  • $\begingroup$ hey and thanks for the tip, i'll definitely remember this approach. however, i'd like to understand the regular way aswell (setting $f(x)=Acos..+Bsin..$) since I won't always think of a good transformation :) $\endgroup$ – armara Feb 26 at 7:49
  • $\begingroup$ You seem to be very focused on the normalization of the variable. This is neither necessary, nor really helpful. It doesn't change the nature of the resolution. $\endgroup$ – Yves Daoust Feb 26 at 8:15
  • $\begingroup$ (after your edit): this makes so much sense... you know that you want either $A=0$ or $B=0$, so you shift both your sine and your cosine to make that happen. and then it's trivial. big thanks, this helped a lot! $\endgroup$ – armara Feb 26 at 8:15
  • $\begingroup$ @YvesDaoust You're right. It was mostly the shifting I wanted to focus on. The scaling just makes it look "nice" $\endgroup$ – Dylan Feb 26 at 8:20

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