Solve dy/dx + 2y = sin x

To solve the differential equation dy/dx + 2y = sin x, we can use the method of integrating factors.

Step 1: Identify the Integrating Factor

The standard form of a first-order linear differential equation is dy/dx + P(x)y = Q(x). Here, P(x) = 2 and Q(x) = sin x.

The integrating factor, μ(x), is calculated as follows:

μ(x) = e^(∫P(x)dx) = e^(∫2dx) = e^(2x)

Step 2: Multiply the Equation by the Integrating Factor

Now, multiply the entire equation by e^(2x):

e^(2x) dy/dx + 2e^(2x)y = e^(2x)sin x

Step 3: Rewrite the Left Side

The left side can be expressed as the derivative of a product:

d/dx (e^(2x)y) = e^(2x)sin x

Step 4: Integrate Both Sides

Next, integrate both sides with respect to x:

∫d/dx (e^(2x)y) dx = ∫e^(2x)sin x dx

This simplifies to:

e^(2x)y = ∫e^(2x)sin x dx + C

To solve the integral on the right, we can use integration by parts or a table of integrals.

Step 5: Solve for y

After finding the integral, isolate y:

y = e^(-2x)(∫e^(2x)sin x dx + C)

Final Solution

The final solution will depend on the result of the integral. You can compute it to get the explicit form of y.