Transform Theory · Engineering Mathematics · GATE CE

The Fourier cosine series of a function is given by :

$$f(x) = \sum\limits_{n = 0}^\infty {{f_n}\cos nx} $$

For f(x) = cos⁴x, the numerical value of (f₄ + f₅) is _________. (round off to three decimal places)

Given two continuous time signals $$x\left( t \right) = {e^{ - t}}$$ and $$y\left( t \right) = {e^{ - 2t}}$$ which exists for $$t>0$$ then the convolution $$z\left( t \right) = x\left( t \right) * y\left( t \right)$$ is ____________.

Laplace transform of $$f\left( x \right) = \cos \,h\left( {ax} \right)$$ is

The Laplace transform of a function $$f(t)$$ is $$$F\left( s \right) = {{5{s^2} + 23s + 6} \over {s\left( {{s^2} + 2s + 2} \right)}}$$$
As $$t \to \propto ,\,\,f\left( t \right)$$ approaches

A delayed unit step function is defined as $$$u\left( {t - a} \right) = \left\{ {\matrix{ {0,} & {t < a} \cr {1,} & {t \ge a} \cr } } \right.$$$

Its Laplace transform is ____________.

If $$L$$ denotes the laplace transform of a function, $$L\left\{ {\sin \,\,at} \right\}$$ will be equal to

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

The Laplace transform of the function
$$\eqalign{ & f\left( t \right) = k,\,0 < t < c \cr & \,\,\,\,\,\,\,\,\, = 0,\,c < t < \infty ,\,\, \cr} $$
is

The Laplace Transform of a unit step function $${u_a}\left( t \right),$$ defined as
$$\matrix{ {{u_a}\left( t \right) = 0} & {for\,\,\,t < a\,} \cr { = 1} & {for\,\,\,t > a,} \cr } $$ is

$${\left( {s + 1} \right)^{ - 2}}$$ is laplace transform of

The inverse Laplace transform of $${{\left( {s + 9} \right)} \over {\left( {{s^2} + 6s + 13} \right)}}$$ is

If $$F\left( s \right) = L\left\{ {f\left( t \right)} \right\} = {{2\left( {s + 1} \right)} \over {{s^2} + 4s + 7}}$$ then the initial and final values of $$f(t)$$ are respectively

Laplace transform of $$f\left( t \right) = \cos \left( {pt + q} \right)$$ is

The Laplace transform of the following function is $$$f\left( t \right) = \left\{ {\matrix{ {\sin t} & {for\,\,0 \le t \le \pi } \cr 0 & {for\,\,t > \pi } \cr } } \right.$$$

Using Laplace transforms, solve $${a \over {{s^2} - {a^2}}}\,\,\left( {{d^2}y/d{t^2}} \right) + 4y = 12t\,\,$$
given that $$y=0$$ and $$dy/dt=9$$ at $$t=0$$

Let $$F\left( s \right) = L\left[ {f\left( t \right)} \right]$$ denote the Laplace transform of the function $$f(t)$$. Which of the following statements is correct?

Using Laplace transform, solve the initial value problem $$9{y^{11}} - 6{y^1} + y = 0$$
$$y\left( 0 \right) = 3$$ and $${y^1}\left( 0 \right) = 1,$$ where prime denotes derivative with respect to $$t.$$