Question

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous. $$ y^{\prime}-\sin t=t^{2} y $$

Short answer

Question: Determine if the differential equation \(y' - (\sin t)y = t^2\) is linear or nonlinear, and if it is homogeneous or nonhomogeneous. Answer: The given differential equation is a linear and nonhomogeneous first-order differential equation.

Step-by-step solution

Understand the definitions of linear, nonlinear, homogeneous, and nonhomogeneous first-order differential equations

A first-order differential equation is in the form: $$ F(y', y, t) = 0 $$ A linear first-order differential equation is in the form: $$ a(t)y' + b(t)y = c(t) $$ where \(a(t)\), \(b(t)\), and \(c(t)\) are continuous functions of \(t\). If the equation is in this form, then it is linear. Otherwise, it is nonlinear. A homogeneous linear first-order differential equation has \(c(t) = 0\). In this case: $$ a(t)y' + b(t)y = 0 $$ A nonhomogeneous linear first-order differential equation has \(c(t) \neq 0\). In this case: $$ a(t)y' + b(t)y = c(t) $$

Analyze the given first-order differential equation

The given equation can be rewritten as: $$ y' - (\sin t)y = t^2 $$ Comparing this with the general form of a linear first-order differential equation, we find that \(a(t) = 1\), \(b(t) = - \sin t\), and \(c(t)=t^2\).

Classify the given equation as linear or nonlinear

Since the given equation is in the form \(a(t)y' + b(t)y = c(t)\), it is a linear first-order differential equation.

Determine if the equation is homogeneous or nonhomogeneous

Since \(c(t) = t^2 \neq 0\), the given equation is a nonhomogeneous linear first-order differential equation.

Key concepts

Linear Differential Equations

When a student starts to explore the world of differential equations, one of the basic distinctions they come across is between linear and nonlinear differential equations. A linear differential equation is one of the simplest forms and is essential to grasp for anybody diving into the subject. Think of it as a straight line in algebra, but instead of variables, you have functions and their derivatives. Linear equations have a standard form \[a(t)y' + b(t)y = c(t)\], where you'll typically see terms involving the function or its derivative, without any multiplications or other operations between them. Basically, if you can rewrite the equation to look like this, keeping in mind that functions like \(a(t)\), \(b(t)\), and \(c(t)\) are just placeholders for any continuous functions of time, you're dealing with a linear differential equation. This structure allows for specific solution methods, like integrating factors or the superposition principle, which can't be used with nonlinear equations.

In our exercise, \(y' - (\text{sin} t)y = t^2\) mimics this form exactly with \(a(t) = 1\) and \(b(t) = -\text{sin} t\) as continuous functions and \(c(t) = t^2\), a non-zero right hand side, confirming it's a linear differential equation. The presence of \(c(t)\) as a non-zero function also tells us something extra about this equation, leading to the next concept.

Nonlinear Differential Equations

On the flip side, we have nonlinear differential equations, which are a bit like the wild west of the differential equation universe. They do not play by the simple rules of linearity, often involving terms where the function and its derivatives are multiplied together, raised to powers other than one, or involved in more complex functions like exponentials or trigonometric functions. Mathematically, a nonlinear equation cannot be molded to fit the linear form \(a(t)y' + b(t)y = c(t)\).

These equations can describe an array range of phenomena, from the chaotic behavior of weather systems to the complex dynamics in biological populations. What makes them tough cookies is that they often require numerical methods for their solutions or, if you're lucky, clever transformations to be soluble analytically. It's also quite common that a single general solution doesn't exist for nonlinear equations, and that's when specific techniques or approximations come into play to at least understand the behavior of solutions. Although our textbook example doesn't showcase a nonlinear differential equation, it's important to keep in mind the versatility and complexity they introduce.

Homogeneous Equations

Imagine a blank canvas, and that's pretty much akin to what homogeneous equations are in the realm of differential equations. In more formal terms, if you're looking at a linear differential equation of the form \(a(t)y' + b(t)y = 0\), you're dealing with a homogeneous equation. The telltale sign is the zero on the right-hand side, which essentially means the equation lacks an external force or influence; the equation depends solely on the function and its derivative.

Homogeneous equations are particularly nice because they have a simpler structure and typically boast straightforward solutions. They often lead to exponential solutions that can be scaled because if \(y(t)\) is a solution, so is any multiple \(ky(t)\). While our example dealt with a nonhomogeneous equation, understanding homogeneous ones lays the groundwork for solving more complex equations and is a fundamental concept in the study of linear differential equations.

Nonhomogeneous Equations

Nonhomogeneous equations bring a sprinkle of diversity into the mix of differential equations. They look similar to their homogeneous cousins but come with an additional non-zero term on the right-hand side, written as \(a(t)y' + b(t)y = c(t)\), where \(c(t)\) is not zero. This term, \(c(t)\), represents an external force or input driving the system described by the equation. So, unlike a homogeneous equation, where solutions typically involve simple functions, nonhomogeneous equations often have more intricate solutions.

These types of equations can be thought of as adding a particular solution (specific to the \(c(t)\) term) to the general solution of the associated homogeneous equation. This mix of solutions quite literally means that nonhomogeneous equations can model systems with an inherent structure that's being acted upon by external forces. Our textbook example, \(y' - (\text{sin} t)y = t^2\), falls into this category, leading us to search for a particular solution that corresponds to the \(t^2\) term while also knowing that we need the general solution of its homogeneous counterpart. This approach opens the door to a rich set of techniques for finding solutions, such as undetermined coefficients or variation of parameters.