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. $$ \frac{d^{2} y}{d x^{2}}+x \sin y=0 $$

Short answer

The given differential equation is an ordinary differential equation of order 2 and is nonlinear.

Step-by-step solution

Identify the type of differential equation

Determine if the given equation is an ordinary or partial differential equation. An ordinary differential equation involves derivatives of the dependent variable with respect to one independent variable. A partial differential equation involves partial derivatives with respect to multiple independent variables. The given equation is: \[ \frac{d^2 y}{d x^2} + x\sin y = 0 \] Since there is only one independent variable (x) and no partial derivatives, this is an ordinary differential equation.

Find the order of the equation

The order of a differential equation is the highest order derivative present in the equation. In our case, the highest order derivative is \(\frac{d^2 y}{d x^2}\), which is a second-order derivative. Therefore, the order of this equation is 2.

Determine if the equation is linear or nonlinear

A linear differential equation is one in which the dependent variable (y in this case) and all its derivatives appear only to the power of one (no higher powers, products, or functions of the dependent variable). If the equation does not meet these criteria, it is considered nonlinear. In our given equation, we have: \[ \frac{d^2 y}{d x^2} + x\sin y = 0 \] The first term, \(\frac{d^2 y}{d x^2}\), is linear, but the second term, \(x\sin y\), makes the equation nonlinear since it involves a function of the dependent variable y, namely the sine function. Hence, the equation is nonlinear. #Summary# To summarize, the given differential equation is an ordinary differential equation, as it involves derivatives with respect to one independent variable. The order of the equation is 2, and it is a nonlinear differential equation.

Key concepts

Differential Equation Classification

Differential equations are crucial tools in mathematics that help describe various dynamic systems. They come in two primary types: ordinary differential equations (ODEs) and partial differential equations (PDEs). Understanding the distinction is essential for solving these equations correctly.

An ordinary differential equation involves functions of one independent variable and its derivatives. For example, in the given equation \( \frac{d^2 y}{d x^2} + x\sin y = 0 \), \( y \) is a function of the variable \( x \), making it an ODE. These are simpler in form because they deal with derivatives concerning only one variable.

On the other hand, partial differential equations involve multiple independent variables and partial derivatives of the function concerning these variables. PDEs are typically more complex due to the involvement of multiple dimensions. In our example, if we had terms like \( \frac{\partial y}{\partial x} \) and \( \frac{\partial y}{\partial t} \), it would be a PDE, but since it is not present, it is confirmed as an ODE.

Differential Equation Order

The order of a differential equation is a simple yet significant concept that determines the equation's complexity and the type of solutions one might expect. It is defined as the highest order of derivative in the equation. In our provided equation \( \frac{d^2 y}{d x^2} + x\sin y = 0 \), the highest derivative present is \( \frac{d^2 y}{d x^2} \).

This derivative is of the second order, making the equation a second-order ODE. Recognizing the order is critical because it often dictates the method you use to find solutions. Low-order differential equations usually have straightforward methods for obtaining solutions, while higher-order equations might require more advanced techniques. Understanding this helps in preparing the necessary strategy to solve the differential equation properly.

Linear and Nonlinear Differential Equations

Differential equations are further classified based on their linearity. This fundamentally affects the methods available for finding their solutions. A linear differential equation allows the dependent variable and all of its derivatives to appear only to the first power. It should not involve products between the function and its derivatives or functions like sine or cosine of the function.

Consider our equation, \( \frac{d^2 y}{d x^2} + x\sin y = 0 \). The term \( \frac{d^2 y}{d x^2} \) is linear. However, the presence of \( x\sin y \) makes the equation nonlinear because \( y \) appears within a sine function. Nonlinear differential equations are often more challenging to solve as they can exhibit complex behavior and may not have closed-form solutions.

Hence, understanding whether a differential equation is linear or nonlinear is essential, as it guides the selection of appropriate solving techniques. While linear equations typically have well-established general solution methods, nonlinear equations often require numerical methods or approximations to find solutions.