The following differential equation is going to be solved using a finiteelement equation:

\[\frac{\mathrm{d}^{2} u}{\mathrm{~d} x^{2}}=0, \quad 1 \leq x \leq 2 \quad u(1)=0, \frac{\mathrm{d} u}{\mathrm{~d} x}(2)=1 \]

Answer the following questions:

a. When one element with two nodes is used, write the expression of approximate solution \(\tilde{u}(x)\) in terms of two nodal values, \(u_{1}\) and \(u_{2}\).

b. Write the two equations of weighted residuals using the Galerkin method. Use the approximate solution obtained from part (a)

c. Solve the equations in part (b) for nodal values, \(u_{1}\) and \(u_{2}\), and approximate solution \(\tilde{u}(x)\).