Question

Find the order and degree of the following differential equations: (i) \(\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\cot \mathrm{x}\). (ii) \(\mathrm{e}^{\left(\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{d}^{3} \mathrm{y}}{\mathrm{dx}^{3}}\right)}=\ln \left(\frac{\mathrm{d}^{5} \mathrm{y}}{\mathrm{dx}^{5}}+1\right)\)

Short answer

Question: Determine the order and degree of the following differential equations: (i) \(\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\cot \mathrm{x}\) (ii) \(\mathrm{e}^{\left(\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{d}^{3}\mathrm{y}}{\mathrm{dx}^{3}}\right)}=\ln \left(\frac{\mathrm{d}^{5}\mathrm{y}}{\mathrm{dx}^{5}}+1\right)\) Answer: (i) The order of the differential equation is 1, and the degree is also 1. (ii) The order of the differential equation is 5, and the degree is 1.

Step-by-step solution

(Order and Degree for Exercise (i))

To find the order and degree of the given equation \(\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\cot \mathrm{x}\), firstly we need to identify the highest order derivative present in the equation. In this case, the highest order derivative is \(\frac{\mathrm{dy}}{\mathrm{dx}}\), which means the order of the differential equation is 1. There is no power raised to the derivative, which means the degree of this equation is 1 as well.

(Order and Degree for Exercise (ii))

To find the order and degree of the given equation \(\mathrm{e}^{\left(\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{d}^{3}\mathrm{y}}{\mathrm{dx}^{3}}\right)}=\ln \left(\frac{\mathrm{d}^{5}\mathrm{y}}{\mathrm{dx}^{5}}+1\right)\), firstly we have to identify the highest order derivative present in the equation. In this case, the highest order derivative is \(\frac{\mathrm{d}^{5}\mathrm{y}}{\mathrm{dx}^{5}}\), which means the order of the differential equation is 5. Now, we need to check the highest power of the highest order derivative present in the equation. To do that, we consider two parts of the equation: the base of exponential function and the content inside the natural logarithm. On the left-hand side, we have an exponential function with exponent \(\left(\frac{\mathrm{dy}}{\mathrm{dx}}-\frac{\mathrm{d}^{3}\mathrm{y}}{\mathrm{dx}^{3}}\right)\). This shows the highest order derivative raised to the power of 1, as no higher powers are present. On the right-hand side, we have \(\ln \left(\frac{\mathrm{d}^{5}\mathrm{y}}{\mathrm{dx}^{5}}+1\right)\). Since the logarithmic function cannot help in identifying the degree for our purposes, we look inside the brackets. The highest order derivative (\(\frac{\mathrm{d}^{5}\mathrm{y}}{\mathrm{dx}^{5}}\)) is raised to the power of 1. As both the parts of the equation have the highest order derivative raised to the power of 1, the degree of this differential equation is 1.

Key concepts

Order of Differential Equation

Understanding the order of a differential equation is fundamental in the subject of differential equations, particularly for students preparing for competitive exams like the IIT JEE. The order is defined as the highest derivative present within the differential equation. For example, consider the differential equation from the exercise, \(\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\cot \mathrm{x}\), where the highest order derivative is \(\frac{\mathrm{dy}}{\mathrm{dx}}\). This indicates that the equation is a first-order differential equation. Grasping this concept allows students to categorize and approach solving the equation with the appropriate method.

It's crucial for students not to confuse the terms 'order' and 'degree', as they are distinct properties of differential equations. The order solely refers to the highest derivative, disregarding any powers or coefficients. When studying or solving differential equations, identifying the order is often the first step and sets the stage for further analysis and solution strategies.

Degree of Differential Equation

Once the order of a differential equation is established, the next concept to understand is the degree of the differential equation. It is defined as the power of the highest order derivative in the differential equation, but only when the equation is a polynomial equation in its derivatives. For instance, the degree of the equation \(\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\cot \mathrm{x}\) is 1, since the first order derivative \(\frac{\mathrm{dy}}{\mathrm{dx}}\) is to the power of 1.

However, it is essential to note that not all differential equations have a degree. Equations that involve terms with trigonometric, logarithmic, or exponential functions of derivatives may not have a defined degree. In the case where these terms can be removed and the equation can be expressed as a polynomial in derivatives, then the degree can be identified. Understanding the degree is pivotal as it can dictate the complexity of solving the equation and the nature of the solutions.

Solving Differential Equations

Solving differential equations is a multi-faceted process and hinges on the order and degree of the equation. For first-order differential equations like \(\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{xy}=\cot \mathrm{x}\), numerous methods such as separation of variables, integrating factor, and substitution can be employed based on the form of the equation. The chosen method should lead to the most straightforward integration and the clearest path to the solution.

To excel in solving differential equations for the IIT JEE, it is essential that students develop a strong understanding of these methods and practice applying them to various types of equations. Furthermore, an important tip for students is to become familiar with the corresponding initial conditions or boundary values, as these conditions help determine the unique solution to the differential equation.

Integral Calculus

Integral calculus plays a significant role in the context of differential equations, particularly when it comes to solving them. The process of integration is often used to find a function given its derivative, which is essentially what a differential equation provides. Thus, integral calculus is the tool by which we can potentially reverse-engineer differential equations to arrive at solutions.

In the IIT JEE context, mastering integral calculus — including techniques such as u-substitution, integration by parts, and partial fraction decomposition — is crucial for students. This is not only because it forms the backbone of solving many differential equations but also because integral calculus itself is a significant portion of the exam's syllabus. An insightful exercise improvement advice for students would be to strengthen their integral calculus skills, which in turn will provide a robust foundation for handling differential equations with confidence and precision.