5–10. First-order linear equations Find the general solution of the following equations.


v'(y) − v/2 = 14

Verified step by step guidance

1. Identify the type of differential equation given: it is a first-order linear ordinary differential equation of the form \(v'(y) + P(y)v = Q(y)\), where \(P(y) = -\frac{1}{2}\) and \(Q(y) = 14\).
2. Find the integrating factor \(\mu(y)\) using the formula \(\mu(y) = e^{\int P(y) \, dy}\). In this case, calculate \(\mu(y) = e^{\int -\frac{1}{2} \, dy}\).
3. Multiply both sides of the differential equation by the integrating factor \(\mu(y)\) to rewrite the left side as the derivative of the product \(\mu(y) v(y)\).
4. Integrate both sides with respect to \(y\) to find \(\mu(y) v(y) = \int \mu(y) Q(y) \, dy + C\), where \(C\) is the constant of integration.
5. Solve for \(v(y)\) by dividing both sides by \(\mu(y)\) to express the general solution explicitly.

Key Concepts

First-Order Linear Differential Equations

These are differential equations of the form y' + P(x)y = Q(x), where the highest derivative is first order and the equation is linear in the unknown function and its derivative. Solving them typically involves finding an integrating factor to simplify the equation.

Integrating Factor Method

This method involves multiplying the entire differential equation by an integrating factor, usually e^(∫P(x)dx), which transforms the left side into the derivative of a product. This allows the equation to be integrated easily to find the general solution.

General Solution of Differential Equations

The general solution includes all possible solutions of a differential equation and typically contains an arbitrary constant. It represents the family of functions satisfying the equation, encompassing both particular and homogeneous solutions.