engineering-mathematics

Let f(t) be a function of t defined for all positive values of t. The Laplace transform of f(t) denoted by L{f(t)} = ∫₀^∞ e^(-st)f(t)dt, provided that the integral exists where s i...

Which one of the options given is the inverse Laplace transform of $\rm \frac{1}{s^3-s}$ ? 𝑢(𝑡) denotes the unit-step function.

The Fourier series expansion of x 3 in the interval −1 ≤ x < 1 with periodic continuation has

The Dirac-delta function ($\delta(t - t_0)$) for $t, t_0 \in \mathbb{R}$, has the following property $\int_a^b \varphi(t)\delta(t - t_0)dt = \begin{cases} \varphi(t_0) & a 0$; $\ma...

The ordinary differential equation $\frac{dy}{dt} = -\pi y$ subject to an initial condition $y(0) = 1$ is solved numerically using the following scheme: $\frac{y(t_{n+1}) - y(t_n)}...

The Laplace Transform of $$f\left( t \right) = {e^{2t}}\sin \left( {5t} \right)\,u\left( t \right)$$ is

If $$P(X)$$ $$ = $$ $$1/4,$$ $$P(Y) = 1/3,$$ and $$\,\,P\left( {X \cap Y} \right) = 1/12,\,\,$$ the value of $$P(Y/X)$$ is

The Laplace transform of $${e^{i5t}}$$ where $$i = \sqrt { - 1} ,$$

Laplace transform of $$\cos \,\left( {\omega t} \right)$$ is $${s \over {{s^2} + {\omega ^2}.}}$$. The Laplace transform of $${e^{ - 2t}}\,\cos \left( {4t} \right)$$ is

The argument of the complex number $${{1 + i} \over {1 - i}},$$ where $$i = \sqrt { - 1} ,$$ is

Consider the differential equation $$\,\,{x^2}{{{d^2}y} \over {d{x^2}}} + x{{dy} \over {dx}} - 4y = 0\,\,\,$$ with the boundary conditions of $$\,\,y\left( 0 \right) = 0\,\,\,$$ an...

The inverse Laplace transform of the function $$F\left( s \right) = {1 \over {s\left( {s + 1} \right)}}$$ is given by

The inverse Laplace transform of $${1 \over {\left( {{s^2} + s} \right)}}$$ is

An analytic function of a complex variable $$z = x + i\,y$$ is expressed as $$f\left( z \right) = u\left( {x,y} \right) + i\,\,v\,\,\left( {x,y} \right)$$ where $$i = \sqrt { - 1}...

If $$F(s)$$ is the Laplace transform of the function $$f(t)$$ then Laplace transform of $$\int\limits_0^t {f\left( x \right)dx} $$ is

If $$f(t)$$ is a finite and continuous Function for $$t \ge 0$$ the laplace transformation is given by $$F = \int\limits_0^\infty {{e^{ - st}}\,\,f\left( t \right)dt,} $$ then for...