Question

If \(\mathrm{m}\) and \(\mathrm{n}\) are order and degree of the equation \(\left(d^{2} y / d x^{2}\right)^{5}+4\left[\left(d^{2} y / d x^{2}\right)^{3} /\left(d^{3} y / d x^{3}\right)\right]+\left(d^{3} y / d x^{3}\right)=x^{2}-1\) then: (A) \(\mathrm{m}=3, \mathrm{n}=2\) (B) \(\mathrm{m}=3, \mathrm{n}=3\) (C) \(\mathrm{m}=3, \mathrm{n}=5\) (D) \(\mathrm{m}=3, \mathrm{n}=1\)

Short answer

The order and degree of the given differential equation are 3 and 1, respectively. Thus, the correct answer is (D) \(\mathrm{m}=3, \mathrm{n}=1\).

Step-by-step solution

Identify the highest-order derivative

The equation contains the following derivative terms: - \(\left(\frac{d^2 y}{dx^2}\right)^5\) - \(\frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}}\) - \(\frac{d^3 y}{dx^3}\) The highest-order derivative in these terms is the third derivative \(\frac{d^3 y}{dx^3}\), as it has the highest number of derivatives.

Identify the highest power of the highest-order derivative

Now we'll examine the highest powers of the third derivative in the equation. In the second and third terms, the third derivative has power 1: - \(\frac{\left(\frac{d^2 y}{dx^2}\right)^3}{\frac{d^3 y}{dx^3}}\) has third derivative with power 1 in the denominator. - \(\frac{d^3 y}{dx^3}\) has third derivative with power 1. Since the highest power of the highest-order derivative is 1, the degree of the differential equation is 1.

Determine the order and degree

We have found that the order of the equation is 3 (due to the presence of third derivative) and the degree is 1 (as the highest power of the highest-order derivative is 1). Therefore, our answer is: (D) \(\mathrm{m}=3, \mathrm{n}=1\)

Key concepts

Differential Equations

Differential equations are a fundamental aspect of mathematics, used to describe phenomena involving changes over time or space. Essentially, these equations involve unknown functions and their derivatives. Differential equations can model real-world systems such as population dynamics, physical phenomena, and engineering problems.

There are different types of differential equations, including ordinary differential equations (ODEs) which involve functions of a single variable and partial differential equations (PDEs) which involve multivariable functions. In the exercise given, the equation presented is an ODE. Within this equation, derivatives represent the rate of change of a function, helping us understand how the function behaves as the variable changes.

• Understanding the behavior of these functions is crucial for solving differential equations.
• Many techniques exist for solving them, such as separation of variables, integrating factors, and numerical methods.

Grasping how to use these derivatives effectively is key to solving and applying differential equations successfully.

Degree of Differential Equation

The degree of a differential equation is often a point of confusion for students. It refers to the highest power of the highest order derivative in the equation when expressed in polynomial form. For clear identification, the equation should not contain roots or powers of the derivatives.

In the exercise, we find the third derivative \(\frac{d^3 y}{dx^3}\) present, but only to the power of 1. This makes the degree of the differential equation equal to 1.

• If derivatives are not already clear in polynomial form, reorganizing or simplifying the equation might be necessary.
• Only the highest order derivative has its power considered in determining the degree.

Understanding this concept helps students more effectively analyze and categorize differential equations, allowing for targeted solution approaches.

Order of Differential Equation

The order of a differential equation relates to the highest order of any derivative present in the equation. It indicates how many times a function is differentiated in the highest order derivative in the equation. This makes it crucial for determining the nature of the solutions to that differential equation.

In the exercise provided, the term \(\frac{d^3 y}{dx^3}\) signifies that the order of the given equation is 3, as it is the highest derivative present. Studying higher order derivatives requires a deep understanding of calculus, specifically the application of differentiation rules.

Recognizing the order is particularly important when choosing a method for solving the equation, as some methods are specifically designed for certain orders of differential equations. Higher order differential equations often model more complex systems, necessitating more advanced methods and greater mathematical understanding.