https://file.aiquickdraw.com/mathbot/file/17266831440406dnz1al9.jpg

Let's solve the problem shown in the image.

We are asked to use **separation of variables** to solve the following initial value problem:

\[
\frac{dy}{dx} = 2x \cos 2x \quad \text{with} \quad y(3) = 0
\]

### Step 1: Separate the variables
Rewriting the equation:

\[
\frac{dy}{dx} = 2x \cos(2x)
\]

Multiply both sides by \(dx\) to separate \(y\) and \(x\):

\[
dy = 2x \cos(2x) \, dx
\]

### Step 2: Integrate both sides
We now integrate both sides. Starting with the right-hand side:

\[
\int dy = \int 2x \cos(2x) \, dx
\]

To solve the integral on the right, we can use **integration by parts**. Let:

- \( u = x \), so \( du = dx \)
- \( dv = 2 \cos(2x) dx \), so \( v = \sin(2x) \)

Using the formula for integration by parts:

\[
\int u \, dv = uv - \int v \, du
\]

We get:

\[
\int 2x \cos(2x) \, dx = 2 \left( x \sin(2x) - \int \sin(2x) \, dx \right)
\]

Now, integrate \(\int \sin(2x) dx\):

\[
\int \sin(2x) \, dx = -\frac{1}{2} \cos(2x)
\]

Thus, the integral becomes:

\[
2 \left( x \sin(2x) + \frac{1}{2} \cos(2x) \right)
\]

Simplifying:

\[
2x \sin(2x) + \cos(2x)
\]

### Step 3: Write the general solution
Now, the equation for \(y(x)\) becomes:

\[
y = 2x \sin(2x) + \cos(2x) + C
\]

### Step 4: Use the initial condition
We are given that \( y(3) = 0 \). Substitute \(x = 3\) and \(y = 0\) to find \(C\):

\[
0 = 2(3) \sin(6) + \cos(6) + C
\]

\[
0 = 6 \sin(6) + \cos(6) + C
\]

Thus,

\[
C = -6 \sin(6) - \cos(6)
\]

### Final solution:

\[
y = 2x \sin(2x) + \cos(2x) - 6 \sin(6) - \cos(6)
\]

Would you like more details on any specific part? Here are some related questions:

1. How does integration by parts work in detail?
2. Why did we choose to use integration by parts in this case?
3. Can we graph this solution for different values of \(x\)?
4. How do we verify the solution with the initial condition?
5. Are there alternative methods to solve this differential equation?

**Tip**: When solving differential equations, always check if integration by parts is the simplest method, or if there's a substitution that might simplify it further.

Use separation of variables to solve the initial value problem: dy/dx = 2x cos(2x), with y(3) = 0.

Learn how to solve the initial value problem dy/dx = 2x cos(2x) using the method of separation of variables. This detailed solution involves integration by parts and applies an initial condition y(3) = 0 to find the constant of integration.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation