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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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
Some clarification is needed. It is not possible in general to write the solution of:Quote:
Originally Posted by perfect
with, in the form
.
To show this just put.
RonL
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
=y(x)-y(0)=\int_{0}^{x}y'(t)dt)
and integrating by parts,
, (1)
since y''=f. Now one integration of the differential equation gives us, and substitute into (1) to get
.
Captainblack:
The function must also satisfy the initial conditions.Quote:
To show this just put...
In fact I was misreading the question statement in another way (the function does not have to satisfy the initial condition but the solution does).Quote:
Captainblack:
The function must also satisfy the initial conditions.Quote:
To show this just put...
RonL