Solve the differential equation dy/dx = 2x + 1.

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Question: Solve the differential equation dy/dx = 2x + 1.

Options:

Correct Answer: y = x^2 + x + C

Solution:

Integrating both sides, we get y = ∫(2x + 1)dx = x^2 + x + C.

Solve the differential equation dy/dx = 2x + 1.

Solve the differential equation dy/dx = 2x + 1.

Step 1: Start with the differential equation dy/dx = 2x + 1.
Step 2: Recognize that dy/dx means we need to find the function y by integrating the right side.
Step 3: Write down the integral we need to solve: y = ∫(2x + 1)dx.
Step 4: Break down the integral: ∫(2x + 1)dx = ∫2xdx + ∫1dx.
Step 5: Solve the first part: ∫2xdx = x^2 (using the power rule of integration).
Step 6: Solve the second part: ∫1dx = x (the integral of a constant is just x).
Step 7: Combine the results from Step 5 and Step 6: y = x^2 + x + C.
Step 8: Remember that C is the constant of integration, which represents any constant value.

Differential Equations – Understanding how to solve first-order differential equations through integration.
Integration – Applying the rules of integration to find the antiderivative of a polynomial function.