Applied Mathematics

Find the tangential and normal components of acceleration of a point describing the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ with uniform speed $\vec{V}$, when the particle is at $(0,\ b)$.

(a) Find the general solution of the given differential equation by variation of parameters. $$3y''-6y'+6y=e^x\sec x$$

(b) Find the power series solution of $(x^2+1)y''+xy'-y=0$

Obtain the Fourier series over the indicated interval for the given function. $$f(x)=\begin{cases} 3\pi+2x, & -\pi<x<0, \\ \pi+2x, & 0<x<\pi \end{cases}$$

(a) Use Newton’s Raphson method to find the solution accurate to within $10^{-4}$ (corrected upto four decimal places) for the given problem. $$x-\cos x=0, \quad [0,\ \pi/2].$$

(b) Solve the system of linear equations using Gauss Seidel method (with three digit rounding arithmetic ) $$3x_1+4x_2-x_3=8$$ $$5x_1+3x_2+2x_3=17$$ $$-x_1+x_2-3x_3=-8$$

(a) Use Euler’s method to approximate the solution of the initial value problem. $$y'=1+y/x, \quad 1\le x\le 2, \quad y(1)=2, \quad \text{with } h=0.25$$

(b) Using Green’s theorem, evaluate $\int_C \vec{F}(\vec{r}).\,d\vec{r}$ counter clock wise around the boundary curve $C$ of the region $R$, where $\vec{F}=\left[\frac{1}{2}xy^4,\ \frac{1}{2}x^4y\right]$, $R$ the rectangle with vertices $(0,\ 0), (3,\ 0), (3,\ 2), (0,\ 2)$.