Hello!

I have problem:

Derive the solution of the ordinary differential equation

d2y/dx2 =f(x),x>0, y(x)=0, dy/dx (0)=0,

in form y(x)= integral from 0 to x [(x-t)f(t)dt].

tnxs

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- August 13th 2007, 08:33 AMperfect[SOLVED] I need help for derive d2y/dx2 in integral form
Hello!

I have problem:

Derive the solution of the ordinary differential equation

**d2y/dx2 =f(x),**x>0, y(x)=0, dy/dx (0)=0,

in form y(x)= integral from 0 to x [(x-t)f(t)dt].

tnxs - August 13th 2007, 10:48 AMCaptainBlack
- August 17th 2007, 12:48 AMRebesques
perfect:

Quote:

Derive the solution of the ordinary differential equation

d2y/dx2 =f(x), x>0, y(x)=0, dy/dx (0)=0,

in form y(x)= integral from 0 to x [(x-t)f(t)dt]

Are you asked to actually prove the formula, or just show this is the solution? Because the latter is easy: Differentiate twice (under the integral sign) to get y"=f, and since y also gives y(0)=0, y'(0)=0, from the uniqueness theorem this is the only solution!

Yes, I know it's cheating, but it's a way. Now if you are asked to derive the formula, write

and integrating by parts,

, (1)

since y''=f. Now one integration of the differential equation gives us , and substitute into (1) to get

.

Captainblack:

Quote:

To show this just put...

- August 17th 2007, 11:49 AMCaptainBlackQuote:

Captainblack:

Quote:

To show this just put...

RonL