Solve the differential equation dy/dx + 2y = 4.

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Question: Solve the differential equation dy/dx + 2y = 4.

Options:

Correct Answer: y = 2 - Ce^(-2x)

Solution:

This is a linear first-order differential equation. The integrating factor is e^(2x). Solving gives y = 2 - Ce^(-2x).

Solve the differential equation dy/dx + 2y = 4.

Solve the differential equation dy/dx + 2y = 4.

Step 1: Identify the differential equation. We have dy/dx + 2y = 4.
Step 2: Recognize that this is a linear first-order differential equation.
Step 3: Find the integrating factor. The integrating factor is e^(∫2dx) = e^(2x).
Step 4: Multiply the entire equation by the integrating factor: e^(2x) * (dy/dx) + 2e^(2x) * y = 4e^(2x).
Step 5: The left side can be rewritten as the derivative of a product: d/dx(e^(2x) * y) = 4e^(2x).
Step 6: Integrate both sides with respect to x: ∫d/dx(e^(2x) * y) dx = ∫4e^(2x) dx.
Step 7: The left side simplifies to e^(2x) * y. For the right side, the integral is 2e^(2x) + C, where C is the constant of integration.
Step 8: Set the two sides equal: e^(2x) * y = 2e^(2x) + C.
Step 9: Solve for y by dividing both sides by e^(2x): y = 2 + Ce^(-2x).

Linear First-Order Differential Equations – These equations can be expressed in the form dy/dx + P(x)y = Q(x) and can be solved using an integrating factor.
Integrating Factor – An integrating factor is a function used to multiply through a differential equation to make it easier to solve.
General Solution – The general solution of a first-order linear differential equation includes a constant of integration, representing a family of solutions.