The Laplace transform of cos^4t is equal to 3/(8s) + s/[2(s2+4)] + s/[8(s2+16)]. So the Laplace formula of cos4t, that is, L{cos4t} is given as follows. L{cos4t} = $\dfrac{3}{8s} + \dfrac{s}{2s^2+8} + \dfrac{s}{8s^2+128}$. Let us now prove the above Laplace transform formula of the fourth power of cost. What is the Laplace of cos4t Answer: … Read more
The Laplace transform of sin^5t is equal to 5/(8s2+8) – 15/(16s2+144) + 5/(16s2+400). So the Laplace formula of sin5t is given as follows. L{sin5t} = $\dfrac{5}{8s^2+8} – \dfrac{15}{16s^2+144} + \dfrac{5}{16s^2+400}$. We now find the Laplace transform of the fifth power of sint. What is the Laplace of sin5t Answer: L{sin5t} = 5/(8s2+8) – 15/(16s2+144) + … Read more
The Laplace transform of sin^4t is denoted by L{sin4t}, and it is equal to 3/(8s) – s/[2(s2+4)] + s/[8(s2+16)]. So the Laplace formula of sin4t is equal to L{sin4t} = $\dfrac{3}{8s} – \dfrac{s}{2(s^2+4)} + \dfrac{s}{8(s^2+16)}$. Let us now prove the above Laplace transform formula of the fourth power of sint. Laplace of sin4t Question: What … Read more
The Laplace transform formula of periodic functions is used to find the Laplace of a periodic function with period T, that is, f(t+T)=f(t). This formula says that the Laplace transform of f(t) is given by L{f(t)} = $\dfrac{\int_0^T e^{-st} f(t) dt}{1-e^{-sT}}$. Laplace Transform of a Periodic Function Theorem: If f: [0, ∞) → ℝ is a … Read more
The Laplace transform of t^2 u(t-1) is equal to e-s[2/s3 + 2/s2 +1/s]. Here we find the Laplace of t2u(t-1) using the second shifting property of Laplace transforms. The formula of the Laplace of t2u(t-1) is given as L{t2u(t-1)} = $e^{-s}\Big[ \dfrac{2}{s^3} + \dfrac{2}{s^2} +\dfrac{1}{s}\Big]$. What is the Laplace of t2u(t-1)? Answer: The Laplace of … Read more
The Laplace transform of u(t-2) is equal to e-2s/s and it is denoted by L{u(t-2)} = e-2s/s. Before we find the Laplace of u(t-2), the shifted unit step function by 2, let us first recall the definition of u(t-2): $u(t-2)= \begin{cases} 0 & \text{ if } t<2 \\ 1 & \text{ if } t \geq … Read more
The Laplace transform of u(t-1) is equal to e-s/s, that is, L{u(t-1)} = e-s/s. Note that u(t-1) is the shifted unit step function by 1 and it is defined as follows. u(t-1) = 0 if t<1 u(t-1) = 1 if t≥1. Let us now learn how to find the Laplace transform of u(t-1). Laplace of … Read more
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
