Question

Classify each of the following differential equations as ordinary or partial differential equations; state the order of each equation; and determine whether the equation under consideration is linear or nonlinear. $$ \left(\frac{d r}{d s}\right)^{3}=\sqrt{\frac{d^{2} r}{d s^{2}}+1} $$

Short answer

The given equation is a second-order, nonlinear Ordinary Differential Equation (ODE).

Step-by-step solution

Classify the equation as ODE or PDE

The equation given is: $$ \left(\frac{dr}{ds}\right)^{3}=\sqrt{\frac{d^{2}r}{ds^{2}}+1} $$ An ODE involves only ordinary derivatives of one or more dependent variables concerning a single independent variable. A PDE involves partial derivatives concerning more than one independent variable. As we observe this equation, the dependent variable is \(r\) and the independent variable is \(s\). All the derivatives in this equation are with respect to only one variable, \(s\). Therefore, the given equation is an Ordinary Differential Equation (ODE).

Determine the order of the equation

To find the order of a differential equation, we identify the highest derivative present in the equation. In our case, the equation contains the following derivatives: 1. \(\frac{dr}{ds}\), which is a first-order derivative. 2. \(\frac{d^{2}r}{ds^{2}}\), which is a second-order derivative. Since there are no higher-order derivatives present, the equation is of order 2.

Classify the equation as linear or nonlinear

To classify the equation as linear or nonlinear, we need to check whether the equation is linear with respect to dependent variables, their derivatives, and the independent variable. A linear ODE can be written in the following general form: $$ a_0(s) \cdot r(s) + a_1(s) \cdot \frac{dr}{ds} + a_2(s) \cdot \frac{d^2r}{ds^2} = g(s) $$ The given differential equation is: $$ \left(\frac{dr}{ds}\right)^{3}=\sqrt{\frac{d^{2}r}{ds^{2}}+1} $$ In this equation, we can see that the first derivative \(\frac{dr}{ds}\) is being raised to the power of 3, and the second derivative term \(\frac{d^2r}{ds^2}\) is under the square root. Both of these terms make the equation nonlinear since it does not follow the general form of a linear ODE. So, the given differential equation is a Nonlinear Ordinary Differential Equation. To sum up: 1. The given equation is an Ordinary Differential Equation (ODE). 2. The equation is of order 2. 3. The equation is nonlinear.

Key concepts

ODE Classification

Understanding the type of a differential equation is crucial for selecting the appropriate method for solving it. Let's start by distinguishing between the main types. An Ordinary Differential Equation, or ODE, involves derivatives of a function concerning only one independent variable. On the other hand, a Partial Differential Equation, or PDE, involves partial derivatives with respect to multiple independent variables.

The exercise provided involves a function given as 'r' dependent on the variable 's', and only ordinary derivatives are present in the equation. There are no partial derivatives as would be evident in a PDE. Hence, as concluded in the step-by-step solution, the equation indeed falls under the category of ODE. It's important to recognize this distinction early on, as it informs the strategy for solving the equation at hand.

Order of Differential Equations

Grasping the concept of order in differential equations is vital for understanding their complexity and behavior. The order of a differential equation is determined by the highest derivative that it contains.

In the exercise provided, the presence of a second-order derivative, denoted as \( \frac{d^{2}r}{ds^{2}} \), signifies that this is a second-order differential equation. No higher derivatives are present, making this classification a relatively straightforward observation. The order is crucial as it often indicates the number of initial conditions needed to solve the equation and gives insight into the behavior of the solution, such as whether it will represent position, velocity, acceleration, etc., in a physical problem.

Linear vs Nonlinear Differential Equations

A pivotal step in analyzing differential equations is distinguishing between linear and nonlinear types. A linear ODE can be expressed in a form where each term is either a constant, the dependent variable, or derivatives of the dependent variable, all to the first power and multiplied by their respective coefficient functions. Any ODE that deviates from this structure is deemed nonlinear.

In our exercise, the equation \( \left(\frac{dr}{ds}\right)^{3} = \sqrt{\frac{d^{2}r}{ds^{2}}+1} \) illustrates a clear deviation from the linear form. With the first derivative raised to the third power and a square root containing the second derivative, these operations violate the criteria for linearity, classifying our ODE as nonlinear. This classification holds strategic implications in solving the equation, as nonlinear ODEs can exhibit a wide range of complex behaviors and often require more sophisticated analytical or numerical methods for their solution.