Use the Laplace transform to solve the given initial-value problem. (d y)/(d t)-y=1, y(0)=0

Name: Problem 35, Chapter 7: A First Course in Differential Equations with Modeling Applications
Uploaded: 2021-11-13T15:23:16-08:00
Duration: 3 min 56 s
Channel: Uma Kumari
Description: Use the Laplace transform to solve the given initial-value problem. (d y)/(d t)-y=1, y(0)=0

step 1 In this question we are required to solve the initial value problem D -Y by D -T minus y is equa

Use the Laplace transform to solve the given initial-value...

Key Concepts

Inverse Laplace Transform

Once the solution is found in the Laplace (frequency) domain, the inverse Laplace transform is used to convert the solution back into the time domain. This process yields the final solution to the original differential equation, expressed in terms of the original time variable.

Transformation of Derivatives

This concept involves applying the Laplace transform to derivatives, where each derivative transforms in a way that incorporates initial conditions. For example, the transform of the first derivative is given by s times the Laplace transform of the function minus the function’s initial value.

Algebraic Manipulation in the Laplace Domain

After applying the Laplace transform to both sides of a differential equation, the problem is converted into an algebraic equation in terms of the Laplace variable. This step involves solving for the transformed function through standard algebraic operations, which simplifies the overall problem.

Laplace Transform

The Laplace transform is an integral transform used to convert functions of time into functions of a complex variable. It is particularly useful in solving differential equations because it transforms differentiation into multiplication by the variable, thereby simplifying the handling of initial-value problems.

Initial Value Problem (IVP)

An initial value problem is a differential equation accompanied by specified values for the function at a given point, usually at time zero. These initial conditions are essential in determining a unique solution for the differential equation.

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