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. $$ y^{\prime}-\sin t=t y^{2} $$

Short answer

Question: Classify the given first-order differential equation and state if it is linear or nonlinear, and if applicable, homogeneous or nonhomogeneous: \(y^{\prime}-\sin t = ty^2\) Answer: The given first-order differential equation is nonlinear. Homogeneous or nonhomogeneous properties do not apply to nonlinear differential equations.

Step-by-step solution

Determine if the differential equation is linear or nonlinear

To determine if the given differential equation is linear or nonlinear, we need to check if the highest power of the dependent variable y (or its derivatives) is greater than 1. In this case, we have the term \(ty^2\). Its highest power is 2 which indicates that the differential equation is nonlinear. So, the given first-order differential equation is nonlinear.

Checking if the equation is homogeneous or nonhomogeneous

Since we already determined the given differential equation is nonlinear in Step 1, there is no need to check for homogeneity or nonhomogeneity, as these properties only apply to linear differential equations.

Key concepts

Linear Differential Equations

When addressing first order differential equations, the concept of linear differential equations is pivotal. These are equations in which the dependent variable, typically denoted as y, and its first derivative, represented as y', appear to the first power and are not multiplied together or composed with functions. In essence, any first order linear differential equation can be represented in the standard form
\[ y' + p(t)y = g(t) \]
where p(t) and g(t) are functions solely of the independent variable, usually t. The simplicity of linear equations lies in their predictable behavior and the variety of methods available to solve them, such as separation of variables, integrating factors, and the use of transformation techniques that rely on linearity.

• Separation of Variables: This method applies if we can express the differential equation as h(y)dy = g(t)dt.
• Integrating Factor: Often used when the equation cannot be neatly separated, this technique involves multiplying through by a carefully chosen function of t that reduces the equation to an integrable form.

Nonlinear Differential Equations

In contrast to their linear counterparts, nonlinear differential equations have at least one term that involves the dependent variable or its derivatives at a power greater than one, or includes a product of them. These equations are more complex due to the broader range of behaviors and solutions they exhibit.
The equation from the original exercise:
\[ y' - \sin(t) = ty^2 \]
is nonlinear because of the term ty^2. This introduces additional complexities in solving techniques and often requires special methods such as power series solutions, perturbation techniques, or numeric methods like Euler's method. It's important to note:

• Solving nonlinear equations might lead to multiple solutions, depending on initial values and conditions.
• They often describe systems with more erratic or diverse behavior, which is crucial in many fields including physics, economics, and biology.

Homogeneous Differential Equations

Homogeneous differential equations are specific to linear differential equations. They take the form
\[ y' + p(t)y = 0 \]
where the function on the right side of the equals sign is zero. The term 'homogeneous' refers to the consistency of the zero degree of the right-hand side function, which implies that the equation is balanced or 'the same throughout.'

One of the key points in solving homogeneous linear differential equations is the existence of a structure that allows the solutions to be combined to form new solutions. For instance, if y1 and y2 are solutions, any linear combination c1y1 + c2y2 is also a solution. This is known as the principle of superposition, a unique trait of homogeneous equations.

• Homogeneous equations are the starting point when finding the general solution to a linear differential equation.
• Techniques like separation of variables and characteristic equations are commonly used to find solutions to these equations.

Nonhomogeneous Differential Equations

Nonhomogeneous differential equations differ from homogeneous ones due to the presence of an additional term that is not zero on the right-hand side of the equation, like:
\[ y' + p(t)y = g(t) \]
where g(t) is a nonzero function. The presence of this term suggests that the system being modeled is influenced by some external force or input. Here's how they stand out:

• They usually represent real-world scenarios where systems have external inputs, forcing functions, or non-zero initial conditions.
• To solve nonhomogeneous equations, techniques such as the method of undetermined coefficients or variation of parameters are often employed.
• The solution to a nonhomogeneous equation is typically the sum of the general solution to its corresponding homogeneous equation and a particular solution to the nonhomogeneous equation.

Understanding the difference between homogeneous and nonhomogeneous can drastically influence the method used for solving the differential equation and interpreting the behavior of the system it represents.