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Reducible to Exact Differential Equations Video Lecture | Mathematics for Competitive Exams
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FAQs on Reducible to Exact Differential Equations Video Lecture - Mathematics for Competitive Exams
| 1. What are exact differential equations? |  |
Ans. Exact differential equations are a type of differential equation where the total differential of a function can be expressed as the sum of the differentials of its variables. In other words, if a differential equation can be written in the form M(x, y)dx + N(x, y)dy = 0, and the partial derivatives of M and N with respect to y and x, respectively, are equal, then it is an exact differential equation.
| 2. How to determine if a given differential equation is exact? | |
Ans. To determine if a given differential equation is exact, you can check if the partial derivatives of the coefficients M and N with respect to their corresponding variables x and y are equal. If Mₓ = Nᵧ, where Mₓ represents the partial derivative of M with respect to x and Nᵧ represents the partial derivative of N with respect to y, then the equation is exact.
| 3. What is the process of solving an exact differential equation? | |
Ans. The process of solving an exact differential equation involves finding a function φ(x, y) such that the total differential of φ is equal to the given equation. This can be done by integrating the equation with respect to x and y separately and then checking if the resulting function satisfies the condition M = φₓ and N = φᵧ. If it does, then the solution is φ(x, y) = C, where C is the constant of integration.
| 4. Can all differential equations be reduced to exact form? | |
Ans. No, not all differential equations can be reduced to exact form. Some differential equations cannot be expressed as the sum of the differentials of its variables, making it impossible to find a function φ(x, y) that satisfies the condition for exactness. In such cases, other methods like integrating factors or substitutions may be used to solve the differential equation.
| 5. Are exact differential equations commonly encountered in real-world applications? | |
Ans. Exact differential equations are commonly encountered in various fields of science and engineering, where they are used to model and solve a wide range of real-world problems. For example, in physics, they are used to describe the behavior of physical systems such as fluid flow, heat transfer, and electrical circuits. In economics, they can be used to model growth rates, supply and demand, and other economic variables. Therefore, understanding and solving exact differential equations is essential for analyzing and predicting various phenomena in the real world.
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Dec 23, 2024 Last updated |
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