Given the differential equation:
\[
\frac{dy}{dx} + y \tan(x) = e^{-0.02x} \cos(x)
\]
we'll solve it using the method of integrating factors.

### Step 1: Write in standard linear form
The given differential equation is already in the standard form:
\[
\frac{dy}{dx} + P(x)y = Q(x)
\]
where \(P(x) = \tan(x)\) and \(Q(x) = e^{-0.02x} \cos(x)\).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:
\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int \tan(x) \, dx}
\]
Calculating the integral of \( \tan(x) \):
\[
\int \tan(x) \, dx = \ln|\sec(x)|
\]
So the integrating factor becomes:
\[
\mu(x) = e^{\ln|\sec(x)|} = \sec(x)
\]

### Step 3: Multiply the differential equation by the integrating factor
Multiplying the entire equation by \( \sec(x) \):
\[
\sec(x) \frac{dy}{dx} + y \sec(x) \tan(x) = e^{-0.02x}
\]
Using the identity \( \sec(x) \tan(x) = \frac{d}{dx}(\sec(x)) \), this equation becomes:
\[
\frac{d}{dx}(y \sec(x)) = e^{-0.02x}
\]

### Step 4: Integrate both sides
Integrating both sides with respect to \(x\):
\[
y \sec(x) = \int e^{-0.02x} \, dx
\]
The integral on the right-hand side is:
\[
\int e^{-0.02x} \, dx = \frac{e^{-0.02x}}{-0.02} = -50e^{-0.02x}
\]
So the equation becomes:
\[
y \sec(x) = -50e^{-0.02x} + C
\]
where \(C\) is the constant of integration.

### Step 5: Solve for \(y\)
\[
y = -50e^{-0.02x} \cos(x) + C\cos(x)
\]

### Final Solution
The solution to the differential equation is:
\[
y = -50e^{-0.02x} \cos(x) + C\cos(x)
\]

Would you like a more detailed explanation of any step, or do you have any questions?

### Related Questions:
1. What are integrating factors, and how do they simplify solving linear differential equations?
2. How can you derive the identity \( \sec(x) \tan(x) = \frac{d}{dx}(\sec(x)) \)?
3. What is the significance of the constant of integration \(C\) in differential equations?
4. How do initial conditions affect the solution of a differential equation?
5. Can integrating factors be used for non-linear differential equations?

### Tip:
Always check if the given differential equation is linear before applying the method of integrating factors. This technique is specifically designed for linear equations.

Explore the solution to the linear differential equation y' + y tan(x) = e^(-0.02x) cos(x) using the method of integrating factors. This detailed breakdown covers key mathematical concepts such as integrating factors and solving first-order linear differential equations. Suitable for advanced high school and undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation