The Laplace transform of 1/t does not exist. In this post, using the gamma function we will prove that the Laplace of 1 over t does not exist. Laplace of 1/t Question: Show that the Laplace transform of 1/t does not exist. Answer: By definition of Laplace transforms, the Laplace of 1/t will be equal … Read more
The Laplace transform of sin^3t is equal to 6/[(s2+1)(s2+9)]. In this article, let us learn to find the Laplace of sin cube t. The Laplace transform formula of sin3t is given below: L{sin3t} = $\dfrac{6}{(s^2+1)(s^2+9)}$. Find the Laplace Transform of sin3t Answer: The Laplace transform of sin cube t is L{sin3t} = 6/[(s2+1)(s2+9)]. Proof: From … Read more
The Laplace transform of t sin2t is equal to L{t sin2t}= 4s/(s2+4)2. In this post, we learn how to find the Laplace of t times sin2t. The Laplace formula of tsin2t is given by L{t sin2t}= $\dfrac{4s}{(s^2+4)^2}$. Find Laplace of t sin2t Question: What is the Laplace of tsin2t? Solution: If L{f(t)}=F(s), then the multiplication … Read more
The Laplace transform of t^2 sin2t is equal to L{t^2 sin 2t}= 4(3s2-4)/(s2+4)3. Here we learn how to find the Laplace of t square sin2t. The Laplace formula of t2sin2t is given by L{t2 sin 2t}= $\dfrac{4(3s^2-4)}{(s^2+4)^3}$. Find Laplace of t2 sin2t Answer: The Laplace of t2 sin2t is 4(3s2-4)/(s2+4)3. Explanation: The Laplace transform of … Read more
The Laplace transform of sint sin2t sin3t is equal to 1/(s2+16)+1/{2(s2+4)} -3/{2(s2+36)}. The Laplace formula of sin t sin 2t sin 3t is given below: L{sint sin2t sin3t} = $\dfrac{1}{s^2+16}+\dfrac{1}{2(s^2+4)}-\dfrac{3}{2(s^2+36)}$. L{sint sin2t sin3t} Question: Find the Laplace Transform of sint sin2t sin3t. Solution: First, we will simplify the expression sint sin2t sin3t. Note that we … Read more
The inverse Laplace transform of s/(s+1) is equal to δ(t)-e-t where δ is the Dirac delta function. The inverse Laplace of s/(s+1) is denoted by L-1{s/(s+1)}, and its formula is given by L-1{s/(s+1)} = δ(t)-e-t. Inverse Laplace of s/(s+1) Question: Find the inverse Laplace of s/(s+1). Solution: Using the partial fraction decomposition, we have the … Read more
The inverse Laplace transform of 1/s(s+1) is 1 -e-t, and it is denoted by L-1{1/s(s+1)}. Here we learn how to find the inverse Laplace of 1 divided by s(s+1). The formula of the inverse Laplace of 1/s(s+1) is given below: L-1{1/s(s+1)} = 1-e-t. Inverse Laplace of 1/s(s+1) Answer: L-1{1/s(s+1)} = 1 -e-t. Explanation: Note that $\dfrac{1}{s(s+1)}$ … Read more
The inverse Laplace transform of 1/s^3 is t2/2. Here we learn how to find the inverse Laplace of 1/s3. The inverse Laplace transform of 1/s3 is denoted by L-1(1/s3) and its formula is given by L-1(1/s3) = t2/2. Inverse Laplace of 1/s3 Answer: The inverse Laplace of 1/s^3 is equal to t2/2. Explanation: We know … Read more
The Laplace transform of the integral of f(t) is given by the following formula L{∫0t f(u) du} = F(s)/s. Here we prove this Laplace formula of integrals. Statement The Laplace transform of an integral is given by $L\Big\{ \int_0^t f(u) du \Big\} = \dfrac{F(s)}{s}$ where L{f(t)} = F(s) denotes the Laplace transform of f(t). Proof … Read more
If the Laplace transform of a function f(t) is known, then the Laplace transform of its derivative f'(t) can be computed. Actually, one can find the Laplace transform of (any finite order) derivatives of f(t). In this article, we will learn how to find the Laplace transform of derivatives and use it to solve an … Read more