Question

Determine the order of the given differential equation; also state whether the equation is linear or nonlinear. $$ \frac{d y}{d t}+t y^{2}=0 $$

Short answer

Answer: The order of the given differential equation is 1, and it is nonlinear.

Step-by-step solution

Determine the order of the differential equation

The given differential equation is: $$ \frac{dy}{dt}+t y^{2}=0 $$ The highest power of the derivative present in this equation is 1, as it only contains the first derivative \(\frac{dy}{dt}\). Therefore, the order of the differential equation is 1.

Check if the equation is linear or nonlinear

A first-order differential equation is linear if it can be written in the form: $$ \frac{dy}{dt} + p(t)y = q(t) $$ Where \(p(t)\) and \(q(t)\) are functions of the independent variable \(t\). The given equation is: $$ \frac{dy}{dt}+t y^{2}=0 $$ Here, we have a quadratic term \(ty^{2}\), and the equation cannot be written in the linear form mentioned above. Therefore, the given differential equation is nonlinear. In summary, the order of the given differential equation is 1, and it is a nonlinear differential equation.

Key concepts

Order of Differential Equation

The order of a differential equation is a fundamental aspect that determines the highest number of derivatives it involves. To ascertain the order, we observe the highest derivative present; this dictates the order of the equation. For example, in the differential equation
\[\frac{dy}{dt}+t y^{2}=0\],
we identify that the highest derivative is \(\frac{dy}{dt}\), a first derivative. Consequently, this tells us that the equation is of first order. It's vital for students to grasp that the order signifies the number of times you have to integrate to obtain the general solution. It can also suggest the complexity of the equation: generally, the greater the order, the more complex the equation can become.

Linear vs Nonlinear Differential Equations

Differential equations can be classified into two broad types: linear and nonlinear. A linear differential equation is one where the dependent variable and all its derivatives appear to the first power and are not multiplied by each other. They possess the standard form
\[\frac{dy}{dt} + p(t)y = q(t)\],
where \(p(t)\) and \(q(t)\) are functions solely of the independent variable. By contrast, a nonlinear differential equation, such as
\[\frac{dy}{dt}+t y^{2}=0\],
includes terms like \(ty^{2}\), which involves a multiplication of the variable \(y\) by itself, thus deviating from the linear form. Nonlinear equations can be much more complex to solve because they may not superimpose, and solutions may not have a closed-form, frequently requiring numerical methods or approximations to find solutions.

First-Order Differential Equation

A first-order differential equation involves only the first derivative of the dependent variable with respect to the independent one. These forms of equations are crucial because they often model naturally occurring phenomena such as growth and decay processes, or simple electrical circuits. The general form is
\[\frac{dy}{dt} = f(t, y)\],
where the function on the right side \(f(t, y)\) can involve the variables \(t\) and \(y\) in various ways. The equation
\[\frac{dy}{dt}+t y^{2}=0\]
is a quintessential first-order differential equation, albeit nonlinear. With first-order equations, an initial condition is usually provided to determine a unique solution, since the general solution will include an arbitrary constant representing an entire family of solutions.