11–16. Initial value problems Solve the following initial value problems.


y'(t) − 3y = 12, y(1) = 4

Verified step by step guidance

1. Identify the type of differential equation: This is a first-order linear differential equation of the form \(y'(t) + p(t)y = q(t)\), where \(p(t) = -3\) and \(q(t) = 12\).
2. Find the integrating factor \(\mu(t)\) using the formula \(\mu(t) = e^{\int p(t) \, dt}\). Here, calculate \(\mu(t) = e^{\int -3 \, dt}\).
3. Multiply both sides of the differential equation by the integrating factor \(\mu(t)\) to rewrite the left side as the derivative of a product: \(\frac{d}{dt}[\mu(t) y(t)] = \mu(t) q(t)\).
4. Integrate both sides with respect to \(t\) to find \(\mu(t) y(t) = \int \mu(t) q(t) \, dt + C\), where \(C\) is the constant of integration.
5. Use the initial condition \(y(1) = 4\) to solve for the constant \(C\), then solve for \(y(t)\) by dividing both sides by \(\mu(t)\).

Key Concepts

First-Order Linear Differential Equations

These are differential equations of the form y' + p(t)y = q(t). They can be solved using an integrating factor, which simplifies the equation into an exact derivative, allowing integration to find the general solution.

Integrating Factor Method

This method involves multiplying the differential equation by an integrating factor, typically e^(∫p(t)dt), to rewrite the left side as the derivative of a product. This facilitates direct integration to solve for y(t).

Initial Value Problems (IVP)

An IVP specifies the value of the solution at a particular point, such as y(1) = 4. After finding the general solution, the initial condition is used to determine the unique constant, yielding a specific solution.