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Chapter 5: Problem 5

Consider the boundary value problem for the deflection of a horizontal beam fixed at one end, $$ \frac{d^{4} y}{d x^{4}}=C, \quad y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(L)=0, \quad y^{\prime \prime \prime}(L)=0 $$ Solve this problem assuming that \(C\) is a constant.

Short Answer

Expert verified
The solution of the given boundary value problem is \(y(x) = \frac{C}{24}x^4\).

Step by step solution

01

Write down the general solution

Consider the homogeneous equation of the differential equation i.e., \(\frac{d^{4} y}{d x^{4}}=0\), it has four linearly independent solutions: \(1\), \(x\), \(x^2\), \(x^3\). Thus the general solution of our differential equation can be written as: \(y(x)=Ax^4 + Bx^3 + Cx^2 + Dx + E\), where A, B, C, D, and E are constants to be determined.
02

Apply the boundary conditions

There are four boundary conditions: \(y(0)=0\), \(y'(0)=0\), \(y''(L)=0\), and \(y'''(L)=0\). By applying these boundaries, we get four equations: \(E = 0\), \(D = 0\), \(2C + 6BL = 0\), and \(6B + 12AL = 0\). Solving these equations help to find the exact form of the solution.
03

Solve for the unknown coefficients

We form all the equations together to get: \(E = 0\), \(D = 0\), \(2C = -6BL\), \(6B + 12AL = 0\). These system of equations gives us: \(B=0\), \(C=0\), \(D=0\), \(E=0\) and \(A= \frac{C}{24}\).
04

Insert the constants back into the general solution

Insert the found values of the constants back into the general solution to obtain: \(y(x) = \frac{C}{24}x^4\).

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