Applied Mathematics

(a) Expand a fourier series of $f(x) = x^2$, $1 < x < 2$

(b) Find equation of integral surface of the differential equation $2y(z-3)p + (2x-z)$ which passes through the circle $x^2 + y^2 = 2x$, $z = 0$.

(a) Solve the equation $4u_{xx} + 5u_{xy} + u_{yy} + u_x + u_y = 2$, also find its canonical form.

(b) Prove that $\int_{-1}^{1} x^n P_n(x)\, dx = \frac{2^{n+1}(n!)^2}{(2n+1)!}$’ where $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^n$ is Legender polynomial of degree $n$.

(a) Solve the system of linear equations by Guass-Seidel iterative method and perform the first three iterations of

$$\begin{aligned} 20x + y - 2z &= 17 \\ 3x + 20y - z &= -18 \\ 2x - 3y + 20z &= 25 \end{aligned}$$