need help with differential equation please

[david]

Solve the fourth order nonhomogenous linear differential equation y⁽⁴⁾( x)=c for 0<x<α subject to the initial conditions: y(0)= 0, y''(0)= 0, y(α)= 0, y''(α)=0. Calculate y(α/2) in terms of α and c.

 

[tkhunny]

Is it easier than you are thinking?

\(\displaystyle y^{(3)}(x) = cx + E\)

\(\displaystyle y"(x) = \frac{c}{2}x^{2} + Ex + D\)

\(\displaystyle y"(0) = 0 \implies D = 0\)

\(\displaystyle y"(x) = \frac{c}{2}x^{2} + Ex\)