Question

Give the order and degree of each equation, and state whether it is an ordinary or partial differential equation. $$3\left(y^{\prime \prime}\right)^{4}-5 y^{\prime}=3 y$$

Short answer

The given differential equation is an ordinary differential equation of second order and fourth degree.

Step-by-step solution

Identifying the Type of Differential Equation

First, determine if the differential equation is ordinary or partial. A partial differential equation involves partial derivatives with respect to more than one independent variable. An ordinary differential equation involves derivatives with respect to a single independent variable only. In this case, since the derivatives are with respect to a single variable (only y' and y'' are present), it is an ordinary differential equation (ODE).

Finding the Order of the Differential Equation

The order of a differential equation is the highest order of derivative that appears in the equation. Look for the highest order of derivative present; in this equation, the highest derivative is the second derivative, denoted by y''. Thus, the order of the equation is 2.

Determining the Degree of the Differential Equation

The degree of a differential equation is the power of the highest order derivative, after the equation has been made rational and polynomial in derivatives. Here, the highest order derivative (y'') is raised to the 4th power. Consequently, the degree of the equation is 4.

Key concepts

Order of Differential Equation

Understanding the order of a differential equation is crucial for students tackling calculus and advanced mathematics. The order fundamentally reflects how many times you would have to differentiate a function to arrive at the equation. Formally, it's defined as the highest order derivative present in the equation. For example, in the equation \(3(y'')^4 - 5y' = 3y\), the presence of the second derivative of y, denoted as \(y''\), indicates that the highest derivative taken is of second order. Hence, this makes the equation a second-order differential equation.
In practice, identifying the order helps in classifying the method one should use to solve the equation and also in understanding the relationship between various derivatives involved.

Degree of Differential Equation

Equally significant is the concept of the degree of a differential equation. After identifying the order, it's important to make the given equation rational and polynomial in its derivatives, which means it should not have any fractional or negative powers, nor should the derivatives be under a radical. Once this form is achieved, the degree is given by the highest power of the highest order derivative. For the equation \(3(y'')^4 - 5y' = 3y\), the highest order derivative, \(y''\), is raised to the power of 4, thus conferring the equation a degree of 4. This is another critical characteristic that often dictates the solution approach and gives insights into the behavior of solutions.

Partial Differential Equation

While the focus of the given example is on ordinary differential equations (ODEs), understanding partial differential equations (PDEs) is also imperative. PDEs involve partial derivatives, which are derivatives with respect to more than one independent variable. They are fundamental in modeling various phenomena involving multidimensional systems, like the diffusion of heat or the propagation of seismic waves. ODEs and PDEs differ in their complexity; whereas ODEs typically describe one-dimensional dynamical systems, PDEs describe multidimensional systems and demand more sophisticated solution techniques. Being versed in the difference between ODEs and PDEs enables a more comprehensive approach to solving real-world problems.

Solution of Differential Equations

Finally, arriving at the solution of differential equations is the ultimate goal. It involves finding a function or a set of functions that satisfy the given differential equation. Solutions can be explicit, where the dependent variable is expressed in terms of the independent ones directly, or implicit, involving a relation between the variables that satisfies the equation. Solving an ODE like \(3(y'')^4 - 5y' = 3y\) often entails finding an integrating factor, employing transformation techniques, or using numerical methods for more complex cases. The solutions provide substantial insights into the behavior of dynamical systems, helping scientists and engineers comprehend how various factors affect the system's evolution over time.
For students, mastering these concepts equips them with the tools to tackle a wide range of problems, from simple mechanical oscillations to the intricate changes in economic markets.