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In this section of the video, the instructor explains the difference between linear and nonlinear differential equations. They illustrate that linear equations have variables and their derivatives raised to the first power without any other operations or functions affecting them. Nonlinear equations, on the other hand, involve variables raised to higher powers or impacted by operations such as trigonometric functions. The instructor also provides examples to help clarify the concepts.

**00:00:00**In this section of the video, the instructor introduces the concept of linear and nonlinear differential equations. He explains that a linear equation is one where the independent and dependent variables are raised to the power of 1 or 0, and there are no operations or functions affecting the variables. On the other hand, a nonlinear equation may have variables raised to higher powers, or affected by operations such as trigonometric functions or multiplication. The instructor then moves on to discuss differential equations, specifically linear ones. He explains that a linear differential equation of nth order is one where the variables are accompanied by a function of x, and gives examples to further illustrate the concept.**00:05:00**In this section, the speaker discusses linear and non-linear differential equations. They explain that linear differential equations are characterized by having the dependent variable, usually represented by the letter "y," and its derivatives raised to the first power only. The coefficients of the dependent variable and its derivatives are also restricted to be multiplied by the independent variable, typically represented by the letter "x." Non-linear differential equations, on the other hand, involve higher powers of the dependent variable and its derivatives or coefficients that do not follow these rules. The speaker provides examples to illustrate these concepts.**00:10:00**In this section, the speaker explains the concept of linearity in differential equations. They provide examples to illustrate when an equation is not linear, such as when the dependent variable or its derivatives are raised to a power, when they are in a denominator, or when they are affected by trigonometric functions, logarithms, or other operations. The speaker emphasizes the importance of recognizing the dependent variable in order to determine linearity, and they provide additional examples for the audience to practice identifying linear and nonlinear equations.**00:15:00**In this section, the speaker discusses the difference between linear and non-linear differential equations. They explain that a linear equation is one where the coefficients do not contain the dependent variable, and the derivatives are not raised to any power. They go through several examples, indicating whether each equation is linear or not by analyzing the coefficients and variables involved. The speaker emphasizes the importance of recognizing the dependent variable in each equation and ensures that viewers understand the distinction between linear and non-linear equations.**00:20:00**In this section, the speaker discusses various differential equations and determines whether they are linear or not. They analyze the presence of derivatives and coefficients in each equation to make their determinations. Through this analysis, they identify which equations are linear and which ones are not. They emphasize the importance of derivatives in differential equations, as well as the accompanying coefficients. Additionally, they mention that functions without the variable "y" can still be linear. The speaker concludes by inviting the audience to explore the complete course for further understanding and to subscribe, comment, and share the video.

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