<s>[INST] Laplace Transform on Beam (w/ Loads) (Sorry about my english I'm a South American student)
The deflection of a fixed beam is zero at $x=0$ and $x=L$ on its boundaries. The beam resists a distributed load of $Wo$ per unit length. The equation is: 
$$\frac{d^{4}y}{dx^4}=\frac{W_{0}}{EI}$$
$$0 < x < L$$
Where $Wo$, $E$, and $I$ are constants
$y(0) = y''(0) = y(L) = y''(L) = 0$
Solve using Laplace Transform 
 [/INST] Take Laplace Transform of the equation, but perform this with respect to x (where normally, it is performed with respect to time, t):
$$ \mathcal{L} \left[ \frac{d^4 y}{dx^4} \right] = \mathcal{L} \left[ \frac{W_0}{EI} \right] $$
$$ s^4 Y(s) - s^3 y(0) - s^2 y^\prime (0) - s y^{\prime \prime}(0) - y^{\prime \prime \prime}(0) = \frac{W_0}{EI} \frac{1}{s} $$
By applying the given boundary conditions, this equation reduces:
$$ s^4 Y(s) - s^2 y^\prime (0) - y^{\prime \prime \prime}(0) = \frac{W_0}{EI} \frac{1}{s} $$
$$ Y(s) = \frac{W_0}{EI} \frac{1}{s^5} + y^{\prime} (0) \frac{1}{s^2} + y^{\prime \prime \prime} (0) \frac{1}{s^4} $$
Now, take inverse Laplace Transform:
$$ \mathcal{L}^{-1} [Y(s)] = \mathcal{L}^{-1} \left[ \frac{W_0}{EI} \frac{1}{s^5} + y^{\prime} (0) \frac{1}{s^2} + y^{\prime \prime \prime} (0) \frac{1}{s^4} \right] $$
$$ y(x) = \frac{W_0}{EI} \frac{x^4}{24} + y^{\prime \prime \prime}(0) \frac{x^3}{6} + y^\prime(0)x $$
$$ y^{\prime \prime}(x) = \frac{W_0}{EI} \frac{x^2}{2} + y^{\prime \prime \prime}(0) x $$
Then, apply other boundary conditions:
$$ y(L) = 0 = \frac{W_0}{EI} \frac{L^4}{24} + y^{\prime \prime \prime}(0) \frac{L^3}{6} + y^\prime(0) L $$
$$ y^{\prime \prime}(L) = 0 = \frac{W_0}{EI} \frac{L^2}{2} + y^{\prime \prime \prime}(0) L $$
which turns out to be
$$ y^{\prime \prime \prime}(0) = - \frac{W_0 L}{2EI} $$
Substitute into first boundary condition:
$$ y(L) = 0 = \frac{W_0 L^4}{24EI} - \frac{W_0 L^4}{12EI} + y^\prime(0) L $$
$$ y^\prime (0) = \frac{W_0 L^3}{24EI} $$
Combining all the terms:
$$ y(x) = \frac{W_0}{24EI} x^4 - \frac{W_0}{12EI} x^3 L + \frac{W_0}{24EI}x L^3 $$
Finally,
$$ y(x) = \frac{W_0}{24EI} \left( x^4 - 2L x^3 + L^3 x \right) $$
which is the deflection of simply supported beam at both sides with distributed loading.
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