The Laplace transform of e^-t sint, that is, L{e-t sint} is equal to 1/[(s+1)2+1]. Note e-t sint is a product of two functions, and its Laplace is calculated using the first shifting property of Laplace transforms. The Laplace formula of e-t sint is given below. L{e-t sint} = 1/[(s+1)2+1]. Laplace of e-t sint Answer: The … Read more
The Laplace transform of (e^at-cosbt)/t is equal to log[(s2+b2)1/2/(s-a)]. Here we will find the Laplace of (eat-cosbt)/t using the division by t formula of Laplace transforms. Note that L{ (eat-cosbt)/t } = log [(s2+b2)1/2/(s-a)]. Laplace of (eat-cosbt)/t To find the Laplace transform of (eat-cosbt)/t, we will follow the steps discussed below. Step 1: At first, … Read more
The Laplace transform of unit step function is equal to 1/s. As the unit step function is denoted by u(t), its Laplace is given by L{u(t)} = 1/s. Before we learn how to find it, let us recall the definition of the unit step function. u(t) = 0 if t<0 u(t) = 1 if t≥0. … Read more
The Laplace Transform of t cosht is equal to (s2+1)/(s2-1)2. The Laplace of tcosh t is denoted by L{t cosht}, and it is given as follows. $\boxed{L\{t \cosh t\} = \dfrac{s^2+1}{(s^2-1)^2}}$ More generally, L{t coshat} = (s2+a2)/(s2-a2)2. Let us now find the Laplace transform of tcosh(at). Find the Laplace of t cosht To find the … Read more
The Laplace Transform of t sinht is equal to 2s/(s2-1)2. The Laplace of the product tsinh t is denoted by L{t sinht}, and its formula is given as follows: $\boxed{L\{t \sinh t\} = \dfrac{2s}{(s^2-1)^2}}$ More generally, L{t sinhat} = 2as/(s2-a2)2. In this post, we will find the Laplace transform of tsinh(at). Laplace of t sinht … Read more
The Laplace transform of sint cost is equal to 1/(s2+4). Here we will learn how to find the Laplace of sint cost. The Laplace of sint cost is denoted by L{sint cost}, and its formula is given by $\boxed{L\{\sin t \cos t\} = \dfrac{1}{s^2+4}}$. Laplace of sint cost Answer: The Laplace of sint cost is … Read more
The Laplace transform of (1-sint)/t does not exist. That is, the formula for the Laplace of 1-sint divided by t is given as follows: L{$\frac{1-\sin t}{t}$} = Undefined. Let us now find the Laplace of (1-sint)/t. What is the Laplace transform of (1-sint)/t? Answer: The Laplace of (1-sint)/t is not defined Solution: To find the … Read more
The change of scale property of Laplace transforms is useful to find the Laplace of f(at) if the Laplace of f(t) is known. It states that if L{f(t)} = F(s) then L{f(at)} = $\dfrac{1}{a} F(\dfrac{s}{a})$. Statement of Change of Scale Property of Laplace Let the Laplace of f(t) be F(s), that is, L{f(t)} = F(s). … Read more
The second shifting property of Laplace transforms is used to find the Laplace of a function multiplied by a unit step function. Let us discuss the second shifting property, its proof and some solved examples. Before we proceed, let us recall the definition of a unit step function. It is defined as follows: u(t-a) = … Read more
The first shifting property of Laplace transforms is used to find the Laplace of a function multiplied by an exponential function. Here we discuss the first shifting property along with its proof and solved examples. State First Shifting Property Statement: If L{f(t)} = F(s), then for s>a we have L{eat f(t)} = F(s-a) Proof: As L{f(t)} … Read more