What is the solution to the first-order linear differential equation dy/dx + y = e^x?
Correct Answer: y = e^x + Ce^(-x)
- Step 1: Identify the differential equation: dy/dx + y = e^x.
- Step 2: Recognize that this is a first-order linear differential equation.
- Step 3: Find the integrating factor, which is e^(∫1 dx) = e^x.
- Step 4: Multiply the entire equation by the integrating factor e^x: e^x * (dy/dx) + e^x * y = e^(2x).
- Step 5: Notice that the left side is the derivative of (y * e^x). So, rewrite it: d/dx(y * e^x) = e^(2x).
- Step 6: Integrate both sides: ∫d/dx(y * e^x) dx = ∫e^(2x) dx.
- Step 7: The left side simplifies to y * e^x, and the right side integrates to (1/2)e^(2x) + C, where C is the constant of integration.
- Step 8: So, we have y * e^x = (1/2)e^(2x) + C.
- Step 9: Solve for y by dividing both sides by e^x: y = (1/2)e^(2x) * e^(-x) + Ce^(-x).
- Step 10: Simplify the expression: y = (1/2)e^x + Ce^(-x).
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