## 2.8.4 Galerkin Method with New Basis

Measurable Outcome 2.12, Measurable Outcome 2.13, Measurable Outcome 2.14

Suppose we introduce another basis function \(\phi _3(x) = x^2(1-x)(1+x)\) into the example problem discussed in Section 2.8.3, which introduces an additional unknown \(a_3\). Use the Galerkin approach to solve for the new values \(a_1\), \(a_2\), and \(a_3\). (Hint: to check your solution, it might be a good idea to plot the estimate of \(\tilde{T}\) to check if it agrees well with the plot above.) Please include at least two decimal places in your answer.

\(a_1\):

\(a_2\):

\(a_3\):

The three residual equations that we solve are: