45–48. General first-order linear equations Consider the general first-order linear equation y'(t)+a(t)y(t)=f(t). This equation can be solved, in principle, by defining the integrating factor p(t)=exp(∫a(t)dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes


p(t)(y′(t) + a(t)y(t)) = d/dt(p(t)y(t)) = p(t)f(t).


Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor.


y′(t) + (2t)/(t² + 1)y(t) = 1 + 3t², y(1) = 4

Verified step by step guidance

1. Identify the given first-order linear differential equation: \(y'(t) + \frac{2t}{t^2 + 1} y(t) = 1 + 3t^2\), with the initial condition \(y(1) = 4\).
2. Determine the integrating factor \(p(t)\) using the formula \(p(t) = \exp\left(\int a(t) \, dt\right)\), where \(a(t) = \frac{2t}{t^2 + 1}\). So, compute \(p(t) = \exp\left(\int \frac{2t}{t^2 + 1} \, dt\right)\).
3. Evaluate the integral inside the exponent to find \(p(t)\). Notice that the integral involves a rational function that can be simplified by substitution.
4. Multiply both sides of the original differential equation by the integrating factor \(p(t)\) to rewrite the left side as the derivative of the product \(p(t) y(t)\): \(\frac{d}{dt} \left(p(t) y(t)\right) = p(t) (1 + 3t^2)\).
5. Integrate both sides with respect to \(t\) to get \(p(t) y(t) = \int p(t) (1 + 3t^2) \, dt + C\). Then solve for \(y(t)\) by dividing both sides by \(p(t)\), and use the initial condition \(y(1) = 4\) to find the constant \(C\).

Key Concepts

First-Order Linear Differential Equations

A first-order linear differential equation has the form y' + a(t)y = f(t), where a(t) and f(t) are functions of t. Such equations describe many physical and mathematical processes and can be solved systematically using integrating factors. Understanding the structure of these equations is essential for applying solution techniques.

Integrating Factor Method

The integrating factor is a function p(t) = exp(∫a(t) dt) used to simplify first-order linear equations. Multiplying the entire differential equation by p(t) transforms the left side into the derivative of p(t)y(t), making it easier to integrate both sides. This method converts the equation into an exact derivative form, facilitating straightforward integration.

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution y(t) at a particular point t = t₀, such as y(1) = 4. Solving an IVP involves finding the general solution of the differential equation and then using the initial condition to determine the unique constant of integration, ensuring a specific solution that fits the given initial data.