Using LAPLACE Transformation to Solve Differential Equations
- Signal and System Spring 2023 Homework Requirements and Reference Answer Summary
- Signals and Systems 2023 (Spring) Assignment Requirements - Eleventh Assignment
- Signals and Systems 2023 (Spring) Homework Reference Answers - 11th Homework
01 The eleventh homework
1. Introduction to Exercises
This is a differential equation, and the unilateral Laplace transform is calculated for it at the same time, and the linear and differential properties of the Laplace transform are applied to convert the differential operation into an algebraic operation. Equations are transformed into algebraic equations. By simplifying it, the Laplace transform of the complete solution of the equation can be obtained. Through the inverse transformation, the time domain expression of the solution of the equation can be obtained. In the eleventh assignment there are two small problems for practicing the solution of differential equations using Laplace transforms.
2. Problem solving
1. The first sub-question
The first sub-problem, is a second order differential equation. Find the Laplace transform on both sides of the equation. For the first term, the second derivative of y(t), write the corresponding Laplace transform, and write the Laplace transform of each term in turn. The input signal u(t) is also subjected to Laplace transform. This yields an algebraic equation for Y(s). Simplify and solve for Y(s). It is factorized to solve the inverse Laplace transform. This is the result of the first sub-question.
2. Second question
The second sub-problem is also a second order differential equation. Find the Laplace transform on both sides, and substitute the initial condition. Solve for Y(s). Next, use the factorization method to solve the inverse transformation of Y(s). One special feature of this small problem is that it has a second-order pole corresponding to s=1. Write down the time domain signal expression corresponding to each term. Combining them together gives the solution of the differential equation. This is the answer to ontology.
※ Summary ※
This paper discusses the application of Laplace transform to solve the exercises of differential equations. Through Laplace transform, the solution of differential equations is greatly simplified.
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