Solve Initial Value Problem Using Laplace Transform | Step-by-Step Guide with Cos(5t)

In this video**, we solve an initial value problem using the **Laplace transformation method**, focusing on a second-order differential equation with a cosine forcing function:
**y'' + 25y = 10cos(5t)**, given initial conditions: **y(0) = 2** and **y'(0) = 0**.

📌 **What you'll learn in this video**:

* How to apply the Laplace transform to solve differential equations
* How to deal with second-order derivatives in Laplace space
* How to handle trigonometric functions like **cos(5t)** using Laplace transforms
* How to find the **inverse Laplace transform**
* How to recognize and use standard Laplace pairs

💡 Whether you're a student preparing for exams or a teacher looking for a better way to explain Laplace transforms, this video is for you.

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### 📘 **Equation Solved**:

y'' + 25y = 10cos(5t), with y(0) = 2, y'(0) = 0

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### ⏱️ Timestamps:

0:00 - Introduction
0:30 - Writing the differential equation
1:10 - Applying Laplace transform to both sides
3:45 - Laplace of second-order derivatives explained
6:30 - Substituting initial conditions
8:15 - Solving for Laplace of Y(s)
10:20 - Finding inverse Laplace transform
13:00 - Final solution
14:30 - Outro and subscription reminder

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### 🎬 Video Chapters:

* **Intro**
* **Step 1: Apply Laplace Transform**
* **Step 2: Use Initial Conditions**
* **Step 3: Solve in Laplace Domain**
* **Step 4: Inverse Laplace Transform**
* **Step 5: Final Answer & Explanation**

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### 📢 Suggested Hashtags:

#laplacetransform #differentialequation #MathTutorial #engineeringmathematics

Видео Solve Initial Value Problem Using Laplace Transform | Step-by-Step Guide with Cos(5t) канала Maths Atoka