Solve the following differential Equation using Galerkin Method.

$\frac{d^2y}{dx^2}+3x \frac{dy}{dx}-6y = 0 \hspace{0.6cm} 0 \lt x \lt 1$

Boundary conditions are : y (0) = 1 , y'(1) = 0.1

Find y (0.2) and compare with exact solution.

Galerkin Method:

order of d.e = 2

Degree of polynomial = 3

Let approximate solution be y

$y = C_0 + C_1x + C_2x^2 + C_3x^3\\ y' = C_1x + 2 C_2x + 3C_3x^2$

From boundary condition 1: y(0) = 1

$C_0 = 1$

From boundary condition 2 : y'(1) …