Advanced Engineering Mathematics I

Show ${\Delta }^3{y}_2={\nabla }^3{y}_5$.

Show that ${\Delta }^2{e}^x=(e-1{)}^2{e}^x$; interval of differencing being unity.

Write the formulae of Trapezoidal rule and simpson 1/3 rule.

Using Euler method find the value of $y(0.025)$. Given $\frac{dy}{dx}=x+y+xy$; $y(0)=1$ and step size $h=0.025$.

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

If $L(F(t))=f(s)$ then find the Laplace Transform of $F^{\prime }(t)$.

Write the formulae of Fourier sine transform and inverse Fourier sine transform.

If $F_c[f(x,t)]=\bar{f}(s,t)$ then write $F_c\left[\frac{{\partial }^{2}f(x,t)}{\partial x^2}\right]$.

State convolution theorem for Z - transform.

Find $Z(n)$.

Show $\Delta {\tan }^{-1}\left(\frac{n-1}{n}\right)={\tan }^{-1}\left(\frac{1}{2n^2}\right)$, interval of differencing being unity.

The distance covered by an athlete for the 50 meter race is given in the following table: | Time (Seconds) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |---|---|---|---|---|---|---|---| | Distance (meters) | 0 | 2.5 | 8.5 | 15.5 | 24.5 | 36.5 | 50 | Determine the speed of the athlete at t = 5 sec.

Find $L(\sin \sqrt{t})$ and hence obtain $L(\cos \sqrt{t}/\sqrt{t})$.

Find ${\int }_0^{\infty }\left(\frac{e^{-t}-e^{-x}}{t}\right)dt$.

Find the Fourier Transform of the following: $$f(x)=\{\begin{array}{ll}1-x^2& |x|\le 1\\ 0& |x|>1\end{array}$$

Solve the following Integral equation: $${\int }_0^{\infty }f(t)\cos atdt=\{\begin{array}{ll}1-a& 0\le a\le 1\\ 0& a>1\end{array}$$

Use convolution theorem to show that $Z^{-1}\left\{\frac{z(z+1)}{(z-1{)}^3}\right\}=n^2$.

From the following table find $f^{\prime }(6)$: | x | 0 | 2 | 3 | 4 | 7 | 9 | |---|---|---|---|---|---|---| | f(x) | 4 | 26 | 58 | 112 | 466 | 922 |

Use Milne's Method to obtain the solution of the equation $\frac{dy}{dx}=x-y^2$ at $x=0.8$ and at $x=1$ given that $y(0)=0$.

Prove $L\left\{\frac{{\sin }^{2}t}{t}\right\}=\frac{1}{4}\log \left(\frac{s^2+4}{s^2}\right)$ and hence deduce the integral ${\int }_0^{\infty }\frac{{\sin }^{2}t}{t^2}dt$.

Prove $e^{-x}\cos x=\frac{2}{\pi }{\int }_0^{\infty }\frac{({\lambda }^2+2)\cos \lambda x}{{\lambda }^4+4}d\lambda$.

Solve the following difference equation by using Z-transform: $$u_{m+2}-u_{m+1}+u_n=n^2{2}^n,u_0=u_1=0$$