Advanced Engineering Mathematics I

Find the Laplace transform of - $$f(t)=\{\begin{array}{ll}\cos t& 0<t<2\pi \\ 0& t>2\pi \end{array}$$

What is unit step function?

Find the Z-transform of sequences - $$\{u_n\}=\{25,10,5,3,2,1,0,5\}-3\le n\le 4$$

Find the inverse Z-transform of $\log \left(\frac{z}{z+1}\right)$ by power series method.

State Convolution Theorem for Fourier transform.

Find the Fourier transform of $f(x)=\{\begin{array}{ll}1& \text{for }|x|<1\\ 0& \text{for }|x|>1\end{array}$.

Prove $E=1+\frac{1}{2}{\delta }^2+\delta \sqrt{1+\frac{{\delta }^2}{4}}$.

By using Lagrange's formula, find x corresponding to y = 10 of following data - | x | 10 | 15 | 17 | 20 | |---|----|----|----|----| | y | 3 | 7 | 11 | 14 |

Find the first approximation value of x by Newton-Raphson method of $f(x)=x{e}^x-2$ upto three decimal places.

Write formula of Milne's Predictor Corrector Method.

Find the inverse Laplace transform of $\frac{2s^2-1}{(s^2+1)(s^2+4)}$.

Find Fourier sine and cosine transform of - $$f(x)=\{\begin{array}{ll}x& \text{for }0<x\le 1\\ 2-x& \text{for }1<x<2\\ 0& \text{for }x\ge 2\end{array}$$

If $\bar{u}(z)=\frac{2z^2+3z+4}{(z-3{)}^3}$, $|z|>3$, then show that $u_1=2$, $u_2=21$ and $u_3=139$.

Evaluate ${\int }_0^1\frac{dx}{1+x^2}$ by using - (i) Trapezoidal rule (ii) Simpson 1/3 rule

Given $\frac{dy}{dx}=x^2+y$, $y(0)=1$. Determine $y(0.02)$ and $y(0.04)$ by using modified Euler's method.

By using Stirling formula, find $u_{32}$ from the following data - $$u_{20}=14.035,u_{25}=13.674,u_{30}=13.257$$ $$u_{35}=12.734,u_{40}=12.089,u_{45}=11.309$$

Solve linear difference equation $u_{n+2}+6u_{n+1}+9u_n=2^n$ given $u_0=1=u_1$.

From the following table of values of x and y find $\frac{dy}{dx}$ and $\frac{d^{2}y}{d{x}^2}$ at $x=1.2$ - | x | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0 | 2.2 | |---|-----|-----|-----|-----|-----|-----|-----| | y | 2.72 | 3.32 | 4.06 | 4.96 | 6.05 | 7.39 | 9.02 |

If $\frac{dy}{dx}=x+y^2$ use Runge-Kutta method to find an approximate value of y for $x=0.2$, given that $y=1$ when $x=0$.

Use Laplace transform to solve - $$(D^2+9)y=\cos 2t,y(o)=1,y(\pi /2)=-1$$

Obtain the Fourier transform of $f(x)=\{\begin{array}{ll}{x}^2& |x|\le a\\ 0& |x|>a\end{array}$. Hence evaluate ${\int }_0^{\infty }\cos \left(\frac{as}{2}\right)\frac{(a^2{s}^2-2)\sin as+2as\cos as}{s^3}ds$.

Find $Z\{a^{|n|}\}$ and hence find $Z\left\{{\left(\frac{1}{2}\right)}^{|n|}\right\}$.