11–18. Solving initial value problems Use the method of your choice to find the solution of the following initial value problems.
y′(t) = -3y + 9, y(0) = 4

Verified step by step guidance

1. Identify the type of differential equation given: \( y'(t) = -3y + 9 \) is a first-order linear ordinary differential equation.
2. Rewrite the equation in standard linear form: \( y' + 3y = 9 \). This helps us recognize the integrating factor method.
3. Calculate the integrating factor \( \mu(t) \) using the formula \( \mu(t) = e^{\int P(t) dt} \), where \( P(t) = 3 \). So, \( \mu(t) = e^{3t} \).
4. Multiply both sides of the differential equation by the integrating factor \( e^{3t} \) to get \( e^{3t} y' + 3 e^{3t} y = 9 e^{3t} \). Notice the left side is the derivative of \( e^{3t} y \).
5. Integrate both sides with respect to \( t \): \( \int \frac{d}{dt} (e^{3t} y) dt = \int 9 e^{3t} dt \). Then solve for \( y(t) \) and apply the initial condition \( y(0) = 4 \) to find the constant of integration.

Key Concepts

First-Order Linear Differential Equations

These are differential equations involving the first derivative of a function and the function itself, typically in the form y' + p(t)y = q(t). Solving them often involves finding an integrating factor or using other methods to obtain the general solution.

Initial Value Problems (IVPs)

An IVP specifies the value of the unknown function at a particular point, such as y(0) = 4. This condition allows us to find a unique solution to the differential equation by determining the constant of integration.

Method of Integrating Factor

This technique solves linear differential equations by multiplying both sides by an integrating factor, usually e^(∫p(t)dt), which simplifies the equation into an exact derivative. Integrating then yields the solution.