1.3.3. Example

Solve:

dy +2y =e−x dx

In this problem, we are tasked with solving a first-order linear differential equation. The equation can be identified by the format of 'dy/dx + P(x)y = Q(x)', where P(x) and Q(x) are functions of x. Here, P(x) is 2 and Q(x) is e^(-x). Our goal is to find the function y that satisfies this equation for all x.

Think of this equation like balancing weights on a seesaw (the left side) against a force that pulls it down (the right side, e^(-x)). The function y is what we want to adjust in order to balance the seesaw at various points.

• P(x)=2, so µ(x)=e^(2dx) =e^(2x)

To solve the differential equation, we first identify the integrating factor, µ(x). The integrating factor is calculated using the formula µ(x) = e^(∫P(x)dx), where P(x) is the coefficient of y in the standard format of the differential equation. For P(x) = 2, we integrate: ∫2dx = 2x, which gives us µ(x) = e^(2x). This factor is crucial as it simplifies the equation, making it easier to solve.

Imagine you are trying to prepare a special drink. The integrating factor is like finding the perfect mix of ingredients (here, e^(2x)) that brings all the flavors together, helping you achieve just the right taste.

• Multiply both sides:

dy d e^(2x) +2e^(2x)y =e^x ⇒ (e^(2x)y)=e^x dx dx

Next, we multiply every term of our original equation by the integrating factor we found, e^(2x). This helps to rewrite the left side of the equation in a specific form, allowing it to be expressed as the derivative of a product: d/dx [µ(x)y]. So, we have d/dx [e^(2x)y] = e^x. This is a key step that transforms our equation to a form that can be easily integrated.

Imagine using a special blender that combines all your ingredients (the left-hand side of the equation) into a smooth mixture, simplifying your next steps in the cooking process, just like how this multiplication simplifies the equation.

• Integrate: ∫(e^(2x)y)dx = ∫e^xdx = e^x + C ⇒ y = e^(-x) + Ce^(-2x)

Now we take the integral of both sides. The left side gives us e^(2x)y, while integrating e^x on the right side yields e^x, plus an arbitrary constant C (since we are looking for a general solution). After integration, we isolate y by dividing by e^(2x) to derive our final solution: y = e^(-x) + Ce^(-2x). This is our general solution to the differential equation.

Think of this final integration as completing a recipe by carefully mixing all the ingredients until you have the final dish (the solution to the equation). You add a pinch of creativity (the constant C) to make it your own.

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