## UnkleRhaukus 2 years ago Verify a solution to the differential equation

1. UnkleRhaukus

2. UnkleRhaukus

$\begin{equation*} \frac{\mathrm d^2 x}{\mathrm dy^2} +k^2y=f(x),\qquad k\in\mathbb R \end{equation*}$

3. UnkleRhaukus

\newcommand\dd[1]{\,\mathrm d#1} % infinitesimal \newcommand\de[2]{\frac{\mathrm d #1}{\mathrm d#2}} % first order derivative \newcommand\pa[2]{\frac{\partial #1}{\partial #2}} % partial derivative \newcommand\den[3]{\frac{\mathrm d^#3 #1}{\mathrm d#2^#3}} % n-th derivative \newcommand\pan[3]{\frac{\partial^#3 #1}{\partial#2^#3}} % n-th partial derivative \begin{align*} y&=\frac1k\int\limits_0^x f(u)\sin k(x-u) \dd u\\ \de yx&=\frac1k\int\limits_0^x \pa {}{x}\Big[f(u)\sin k(x-u)\Big] \dd u\\ &=\frac1k\int\limits_0^x f_x(u)\sin k(x-u)+kf(u)\cos k(x-u) \dd u\\ \\ \\ \den yx2&=\frac1k\int\limits_0^x \pa {}x\Big[f_x(u)\sin k(x-u)+kf(u)\cos k(x-u)\Big] \dd u\\ &=\frac1k\int\limits_0^x f_{xx}(u)\sin k(x-u)+kf_x(u)\cos k(x-u)\\&\qquad\qquad+kf_x(u)\cos k(x-u)-kf(u)\sin k(x-u)\dd u\\ &=\frac1k\int\limits_0^x\Big( f_{xx}(u)-kf(u)\Big)\sin k(x-u)+2kf_x(u)\cos k(x-u)\dd u\\ \end{align*}

4. UnkleRhaukus

\newcommand\dd[1]{\,\mathrm d#1} % infinitesimal \newcommand\de[2]{\frac{\mathrm d #1}{\mathrm d#2}} % first order derivative \newcommand\pa[2]{\frac{\partial #1}{\partial #2}} % partial derivative \newcommand\den[3]{\frac{\mathrm d^#3 #1}{\mathrm d#2^#3}} % n-th derivative \newcommand\pan[3]{\frac{\partial^#3 #1}{\partial#2^#3}} % n-th partial derivative {\small\begin{align*} \den xy2 +k^2y =f(x)\\ \frac1k\int\limits_0^x\Big( f_{xx}(u)-kf(u)\Big)\sin k(x-u)+2kf_x(u)\cos k(x-u)\dd u\\+k\int\limits_0^x f(u)\sin k(x-u) \dd u=f(x)\\ \int\limits_0^x\Big( \frac{f_{xx}(u)}k-f(u)\Big)\sin k(x-u)+2f_x(u)\cos k(x-u)\\+ kf(u)\sin k(x-u) \dd u=f(x) \\ \int\limits_0^x\left( \frac{f_{xx}(u)}k+(k-1)f(u)\right)\sin k(x-u)+2f_x(u)\cos k(x-u) \dd u=f(x) \\ \end{align*}}

5. UnkleRhaukus

now i guess i just have to apply the product rule in reverse ,

6. AccessDenied

Hmm, would that differentiation under the integral be applicable if the domain of integration also contains the variable we are differentiating with respect to? I'm not exactly certain of this...

7. UnkleRhaukus

im not sure

8. wio

Why are you pulling that partial derivative in? Isn't it implied by fundamental theorem that the function inside the integral is the derivative?

9. wio

Hmmm oh yeah, there is that $$x$$ in there...

10. UnkleRhaukus

i dunno

11. wio

What inspired the partial derivative, was it just a guess?

12. AccessDenied

I believe that is an integration rule being used here: http://en.wikipedia.org/wiki/Leibniz_integral_rule

13. UnkleRhaukus

@wio im not sure, it worked on the previous question

14. wio

Ah, I see. I was a big confused as to the fact that we have a multivariable function

15. AccessDenied

Actually, the part titled "Formal Statement" on that page looks useful here...

16. wio

What would $$b(\theta)$$ ad $$\theta$$ be?

17. wio

$$b(\theta) = x$$ and $$\theta = x$$?

18. wio

@UnkleRhaukus did you look at the formal statement?

19. UnkleRhaukus

yeah , but im not sure how to use that in my problem

20. wio

Did you try writing it out to see what happens?

21. AccessDenied

$$\displaystyle \frac{\text{d}}{\text{d} x} \int_{a(x)}^{b(x)} y(u, \; x) \; \text{d}u = \int_{a(x)}^{b(x)} y_x (u, \; x) \; \text{d}u + y(b(x), \; x) \; b'(x) - y(a(x), \; x) \; a'(x)$$ $$a(x) = 0$$, $$b(x) = x$$, $$y(u, \; x) = f(u) \; \sin(k(x - u))$$ I think, anyways. I haven't used this theorem before... (:

22. wio

@AccessDenied actually it seems when you write it out, the added parts go to 0

23. AccessDenied

Yeah, I see that now. Interesting. :)

24. wio

Ummm does $\sin k(x-u) = \sin(kx)\cos(ku) - \sin(ku)\cos(kx)$make things any easier? grasping at straws here.

25. wio

You know one thing I've been wondering about.... Since $$f(u)$$ is obviously a single variable function, does it have a partial derivative?

26. wio

Either it's constant with respect to $$x$$ or $$\frac{d}{dx}f(u) = f'(u) u'(x)$$ based on chain rule.

27. wio

@UnkleRhaukus @AccessDenied could you clear this up?

28. wio

How can a single variable function have a partial derivative?

29. wio

imagine if $$f(x) = e^x$$ the $$f(u)=e^u$$ right?

30. UnkleRhaukus

um

31. wio

I think $$f_x(u)$$ is a dead end. I don't see how you'd recover from that because there is no integration with respect to $$x$$.

32. wio

And it doesn't really make sense to me anyway, for whatever that is worth. Maybe it would work if you just treat $$f(u)$$ like a constant?

33. wio

@UnkleRhaukus Ummm sorry if I'm being a bother but.... do we have reason to believe that $$u$$ and $$x$$ depend upon each other or are independent of each other?

34. UnkleRhaukus

im not sure

35. wio

If you do this: $\large \frac{dy}{dx} = \int^x_0 f(u)\cos k(x-u) du$

36. wio

then try the thing that @AccessDenied mentioned.. I wonder

37. wio

$\large \frac{d^2y}{dx^2} = -k\int^x_0 f(u)\sin k(x-u) du +f(x)$ Oh pellet, it works!

38. wio

@UnkleRhaukus All you gotta do is treat $$f(u)$$ as a constant and it'll work.

39. UnkleRhaukus

i dont understand

40. AccessDenied

Ahh, sorry, I tried to switch to my other computer but I was getting blank screen for the rest of the night. >.< It does appear to work for us here by treating f(u) as independent with respect to x. We were also not told how they were related, so I guess its just more reasonable to assume they are independent of each other.

41. AccessDenied

From the previous problem, we had constant limits of integration. The theorem here 'worked' but the terms involving the derivative of the constants vanished. Here, it becomes a bit different where a limit of integration is a nonconstant function of the differentiating variable and has a nonzero derivative. Since they're both the same theorem though, it may be helpful to keep the general theorem in mind and just understand the result of using constant limits when you see them. That's what I am thinking, anyways.