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.