Question

Find the order of the equation. (a) \(\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x} \frac{d^{3} y}{d x^{3}}+x=0\) (b) \(y^{\prime \prime}-3 y^{\prime}+2 y=x^{7}\) (c) \(y^{\prime}-y^{7}=0\) (d) \(y^{\prime \prime} y-\left(y^{\prime}\right)^{2}=2\)

Short answer

1. \(\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x} \frac{d^{3} y}{d x^{3}}+x=0\) 2. \(y^{\prime \prime}-3 y^{\prime}+2 y=x^{7}\) 3. \(y^{\prime}-y^{7}=0\) 4. \(y^{\prime \prime} y-\left(y^{\prime}\right)^{2}=2\) Answer: The orders of the given differential equations are: 1. Order 3 2. Order 2 3. Order 1 4. Order 2

Step-by-step solution

Find the order of equation (a)

The highest derivative in equation (a), \(\frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x} \frac{d^{3} y}{d x^{3}}+x=0\), is \(\frac{d^{3} y}{d x^{3}}\). Thus, the order of equation (a) is 3.

Find the order of equation (b)

The highest derivative in equation (b), \(y^{\prime \prime}-3 y^{\prime}+2 y=x^{7}\), is \(y^{\prime \prime}\). Thus, the order of equation (b) is 2.

Find the order of equation (c)

The highest derivative in equation (c), \(y^{\prime}-y^{7}=0\), is \(y^{\prime}\). Thus, the order of equation (c) is 1.

Find the order of equation (d)

The highest derivative in equation (d), \(y^{\prime \prime} y-\left(y^{\prime}\right)^{2}=2\), is \(y^{\prime \prime}\). Thus, the order of equation (d) is 2.

Key concepts

Differential Equations

Differential equations are a type of equation that involve derivatives, which represent rates of change. These equations are fundamental in expressing the dynamics of various physical systems, such as the motion of particles, the flow of current in electrical circuits, and the changes in populations over time.

To solve a differential equation means to find a function, or a set of functions, that satisfies the given equation when the derivatives are taken with respect to one or more variables. These solutions often describe a system's behavior and can vary from simple expressions to complex functions depending on the nature of the equation.

For instance, when examining the simple harmonic motion of a pendulum, a second-order differential equation is used to describe its displacement over time. This highlights the broad applications of differential equations across both the natural and social sciences.

Highest Derivative

The Significance of the Highest Derivative:
The 'order' of a differential equation is determined by the highest derivative that appears in the equation. The order gives us a clue about the complexity of the solution and the number of initial conditions needed to solve the equation uniquely.

For example, if you have the highest derivative as the second derivative \( y'' \), it indicates a second-order differential equation. Solving this would typically require two initial values, often corresponding to an initial position and initial velocity in physical contexts.

The exercises provided demonstrate this concept, where various differential equations have their order determined by identifying the term with the highest derivative. This information sets the stage for further analysis and solution-finding techniques tailored to the order of the equation.

Mathematical Notation

Understanding Mathematical Symbols:
Mathematical notation serves as a concise and precise language for mathematicians and scientists. It allows them to write complex ideas in a compact form. Different notations signifying derivatives include Leibnitz's notation \( \frac{dy}{dx} \), the prime notation \( y' \), and the derivative operator \( D \).

In the context of our exercise examples, the prime notation (as in \( y' \) or \( y'' \)) is used to denote the first and second derivatives of a function with respect to a single variable, typically time or space. The power of mathematics comes alive when such a simple symbol can convey so much about the behavior of dynamic systems, making it a tool of immense power for students to grasp.