Solving y' + 2y = e^x

Differential Equations · Axiom Academy

Master the integrating factor method through a complete step-by-step solution. Find the general solution of the linear first-order differential equation y' + 2y = e^x . This equation is already written in standard form y' + P(x)y = Q(x) , so we can solve it directly with the integrating factor method. Excellent work! You've solved a first-order linear differential equation using the integrating factor method. Here's what carries forward: Standard form is essential: always write the equation as y' + P(x)y = Q(x) before identifying P(x) . Integrating factor formula: transforms the left side into a perfect derivative — no constant of integration is needed when finding . The magic of perfect derivatives: after multiplying by , the left side becomes , which is the key insight of this method. Integration completes the solution: once you have the perfect-derivative form, integrate both sides and divide by to solve for y . Don't forget the constant: the constant C represents the entire family of solutions to the differential equation, not just one curve. The integrating factor method is one of the most powerful techniques for solving first-order linear ODEs. Practice recognizing when to use it, and the pattern will become second nature.

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