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Homogeneous Differential Equations and their Solution Video Lecture | Mathematics (Maths) Class 12 - JEE
FAQs on Homogeneous Differential Equations and their Solution Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What is a homogeneous differential equation? |  |
A homogeneous differential equation is a type of differential equation in which all the terms involving the dependent variable and its derivatives are of the same degree. In other words, the equation is homogeneous if it can be written in the form F(x, y, dy/dx) = 0, where F is a function that satisfies the property F(tx, ty, t(dy/dx)) = 0 for any non-zero constant t.
| 2. How do you solve a homogeneous differential equation? | |
To solve a homogeneous differential equation, you can use the technique of separation of variables. First, rewrite the equation in the form dy/dx = f(x, y), where f(x, y) is a function that can be expressed as a ratio of two homogeneous functions of the same degree. Then, separate the variables by putting all the terms involving y on one side and all the terms involving x on the other side. Integrate both sides of the equation to find the general solution.
| 3. Can all differential equations be solved using the method of separation of variables? | |
No, not all differential equations can be solved using the method of separation of variables. The method of separation of variables can only be applied to first-order ordinary differential equations that are separable. Homogeneous differential equations are a special case that can be solved using separation of variables, but other types of differential equations may require different methods such as integrating factors, substitution, or using specific formulas.
| 4. What is the significance of the term "homogeneous" in homogeneous differential equations? | |
In the context of differential equations, the term "homogeneous" refers to the property that all the terms in the equation have the same degree. This property allows for a simplification of the equation and enables the use of certain techniques, such as separation of variables, to find the solution. It is important to identify whether a differential equation is homogeneous or not in order to determine the appropriate method for solving it.
| 5. Can a homogeneous differential equation have non-zero constant solutions? | |
Yes, a homogeneous differential equation can have non-zero constant solutions. In fact, constant functions are always solutions to homogeneous differential equations. This is because when you substitute a constant value for the dependent variable and its derivatives in a homogeneous equation, all the terms vanish and the equation holds true. However, it is important to note that constant solutions are not the only solutions, and there may exist other non-constant solutions as well.
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