In Exercises 5–8, show that each function is a solution of the given initial value problem.
5. Differential Equation: 2y + y' = 4x + 2
Initial condition: y(-1) = e² - 2
Solution candidate: y = e^(-2x) + 2x

Verified step by step guidance

1. Identify the given differential equation: \(2y + y' = 4x + 2\) and the initial condition \(y(-1) = e^{2} - 2\).
2. Calculate the derivative of the candidate solution \(y = e^{-2x} + 2x\). Use the chain rule for the exponential term: \(y' = \frac{d}{dx}(e^{-2x}) + \frac{d}{dx}(2x) = -2e^{-2x} + 2\).
3. Substitute the candidate solution \(y\) and its derivative \(y'\) into the left side of the differential equation: \(2y + y' = 2(e^{-2x} + 2x) + (-2e^{-2x} + 2)\).
4. Simplify the expression from the substitution to verify if it equals the right side of the differential equation \(4x + 2\). This confirms whether the candidate function satisfies the differential equation.
5. Check the initial condition by substituting \(x = -1\) into the candidate solution \(y = e^{-2(-1)} + 2(-1)\) and verify if it equals \(e^{2} - 2\). This confirms the solution meets the initial value condition.

Key Concepts

Initial Value Problem (IVP)

An initial value problem consists of a differential equation paired with a specific condition that the solution must satisfy at a given point. This condition, called the initial condition, ensures a unique solution by specifying the value of the function at a particular input, such as y(-1) = e² - 2.

Verification of a Solution to a Differential Equation

To verify a candidate solution, substitute it and its derivative into the differential equation to check if the equation holds true for all x in the domain. This process confirms whether the proposed function satisfies the relationship defined by the differential equation.

Differentiation of Exponential and Polynomial Functions

Differentiation rules for exponential functions like e^(-2x) and polynomial terms such as 2x are essential to find y'. The derivative of e^(-2x) is -2e^(-2x), and the derivative of 2x is 2, which are used to substitute into the differential equation for verification.