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:
$\displaystyle
\frac{d^2y}{dx^2}=f(x)
$
with $\displaystyle y(0)=0,\ y'(0)=0$, in the form
$\displaystyle y(x)=\int_0^x (x-t) f(t) dt$.
To show this just put $\displaystyle f(x)=x$.
RonL
perfect:
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
$\displaystyle y(x)=y(x)-y(0)=\int_{0}^{x}y'(t)dt$
and integrating by parts,
$\displaystyle y(x)=\int_{0}^{x}y'(t)(t)'dt=y'(t)t\bigg|_{0}^x-\int_0^xy''(t)tdt=xy'(x)-\int_0^xf(t)tdt$, (1)
since y''=f. Now one integration of the differential equation gives us $\displaystyle y'(x)=\int_0^xf(t)dt$, and substitute into (1) to get
$\displaystyle y(x)=x\int_0^xf(t)dt-\int_0^xf(t)tdt=\int_0^x(x-t)f(t)dt$.
Captainblack:
The function must also satisfy the initial conditions.To show this just put...