Laplace Transforms - Examples & Applications

Direct Laplace Transforms

Direct Laplace Transform (Polynomial and Exponential)

Find the Laplace Transform of f(t)=3t2−e−4t+5f(t) = 3t^2 - e^{-4t} + 5.

Direct Transform Solution

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Direct Laplace Transform (First Shift Theorem)

Find the Laplace Transform of f(t)=e2tsin⁡(3t)f(t) = e^{2t}\sin(3t).

First Shift Theorem Solution

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Inverse Laplace Transforms

Inverse Transform (Completing the Square)

Find the inverse Laplace transform of:
F(s)=2s+5s2+6s+34F(s) = \frac{2s+5}{s^2+6s+34}

Completing the Square Solution

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Inverse Transform (Partial Fractions with Repeated Roots)

Find the inverse Laplace Transform of:
F(s)=s2+9s+2(s−1)2(s+3)F(s) = \frac{s^2 + 9s + 2}{(s-1)^2(s+3)}

Partial Fractions Solution

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Solving Initial Value Problems

IVP Problem (First-Order)

Solve the initial value problem using Laplace Transforms:
y′+3y=13sin⁡(2t),y(0)=6y' + 3y = 13\sin(2t), \quad y(0) = 6

First-Order IVP Solution

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IVP Problem (Second-Order)

Solve the initial value problem:
y′′−y′−2y=0,y(0)=1,y′(0)=2y'' - y' - 2y = 0, \quad y(0) = 1, \quad y'(0) = 2

Second-Order IVP Solution

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Step Functions and Dirac Delta

Step Function Transform

Find the Laplace Transform of f(t)=(t−1)U(t−1)f(t) = (t-1)\mathcal{U}(t-1).

Step Function Solution

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Inverse Step Function Transform

Find the inverse Laplace transform of F(s)=e−2ss2+9F(s) = \frac{e^{-2s}}{s^2 + 9}.

Inverse Step Function Solution

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Dirac Delta Problem

Solve the initial value problem representing an undamped mass-spring system struck by a hammer at t=πt = \pi:
y′′+y=3δ(t−π),y(0)=1,y′(0)=0y'' + y = 3\delta(t - \pi), \quad y(0) = 1, \quad y'(0) = 0

Dirac Delta IVP Solution

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Convolution Theorem

Convolution Theorem Problem 1

Find the inverse Laplace transform of H(s)=1s(s2+1)H(s) = \frac{1}{s(s^2+1)} using the Convolution Theorem.

Convolution Theorem Solution 1

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Convolution Theorem Problem 2

Evaluate the convolution f∗gf * g directly, given f(t)=etf(t) = e^t and g(t)=e2tg(t) = e^{2t}.

Convolution Direct Evaluation

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