Professor-Leonard
This video discusses the method of solving separable differential equations, which is the most fundamental technique in solving differential equations. The process involves separating the variables, with the Y functions and dy on one side and the X functions and dx on the other side. Both sides are then integrated, and the resulting function has an arbitrary constant, making it a family of curves. The video provides several examples and emphasizes the importance of not forgetting the arbitrary constant when solving for y. The speaker also demonstrates the use of integration techniques and inverse functions to simplify the solution. The video concludes with tips for simplifying equations and the properties of logarithms.
In this section, the instructor introduces the technique of solving differential equations called separable differential equations, which is the most fundamental and straightforward technique. He explains that we need to group all of the Y functions and dy on one side and all of the X constants and DX on the other side. By doing so, we can integrate both sides and solve for Y explicitly. The instructor also explains that this technique works for any differential equation where we can solve for dy/dx, and the X and Y need to be multiplied together to allow for easy separation of variables. He further adds that he will be covering other techniques in the following videos and will provide a thought process on how to approach differential equations.
In this section, the instructor is discussing separable differential equations and how to solve them. If the differential equation can be written as dy/dx equal to a function of x times a function of y, then the variables can be separated by putting all y terms on one side and all x terms on the other side. When integrating, it is important to keep all constants on the x side to make solving for y easier. The instructor provides several examples of this technique, encouraging viewers to skip ahead if they understand the concept. The video also covers singular solutions that cannot be obtained through integration alone. The first example involves moving variables to separate the equation, and then integrating to solve for y.
In this section, the video discusses the fundamental idea of solving equations by performing operations on both sides and how integrals can be used as an operation. Separable differential equations involve separating the variables and making them look like a product, integrating both sides, and solving for y. The video emphasizes that only one arbitrary constant is needed, which can be added or subtracted from either side to create a general solution. The video also explains how the variables and differentials can be integrated with respect to different variables and cancels out, and that the constant is denoted as C sub 1 to differentiate it from any other arbitrary constant that may be added later.
In this section of the video, the instructor explains how to solve separable differential equations and the concept of the arbitrary constant. The process involves taking e on both sides and treating both of these as an exponent. The inverse of exponential is natural logarithm (Ln). The absolute value needs to be subtracted to solve the equation explicitly. The video also explains the commonly used ideas in solving these types of equations like addition and multiplcation in exponents and the definition of absolute value.
In this section, the concept of an arbitrary constant in separable differential equations is discussed. The constant is arbitrary because it could be any number, and it doesn't have any specific meaning without knowing the specific problem. The derivative of the arbitrary constant is going to satisfy the original differential equation, and any arbitrary constant is going to work. Therefore, we can create a general solution that includes the arbitrary constant. Later on, when we have the initial value, we can solve for the particular solution by using a specific value for the arbitrary constant.
In this section, the speaker discusses separable differential equations and solving for y as a general solution. The first step is to see if the function fits the model of writing dy/dx as a function of x times a function of y. If it does, one can divide to get the ys on one side and the dx on the other side, integrate both sides, and end up with a general solution that satisfies the differential equation. The resulting function has an arbitrary constant, making it a family of curves. The speaker provides examples and emphasizes the rule of integrating one over y, the abundance of absolute value ys, and the plus or minus e to the c1 that often occurs in basic differential equations.
In this section, the video explains how to integrate separable differential equations through examples. The video starts by introducing an example with the integral of sine x, and shows how to solve it by remembering some integration techniques. Then, the video goes on to explain how to solve for y, after integrating both sides. The next example shows how to divide and multiply to integrate both sides, with an emphasis on pulling constant factors. Overall, the video showcases how to solve differentiation equations in a structured way, which involves simple algebra and calculus techniques.
In this section, the video discusses separable differential equations and how to solve them with an example problem. The transcript goes through the steps of moving X to one side and keeping ln(y) on the other, integrating, and then moving coefficients for ln. The video also notes that any coefficient in front of the natural log acts as an exponent. The transcript points out a mistake made during the problem and goes over how to correctly solve it. In the end, the general solution for the differential equation is found by removing the absolute value and adding an arbitrary constant.
In this section, the speaker discusses the process of solving separable differential equations and demonstrates the application of integration techniques. They introduce the concept of grouping the X's and Y's on opposite sides and use examples to illustrate how to rearrange the equation and integrate to solve for Y explicitly. The speaker also emphasizes the importance of not forgetting the arbitrary constant when solving for Y and demonstrates how to use inverse trig functions to express the solution in a simplified form. They conclude with additional examples to reinforce the concept and encourage viewers to review calculus concepts if they are having difficulty understanding the integration techniques involved.
In this section, the speaker discusses how to solve separable differential equations. They provide an example of a differential equation with a function of X and a function of Y that is multiplied together. By rearranging the equation and moving the variables to opposite sides, they are able to integrate and solve for Y. The speaker emphasizes not overthinking the process and provides tips for simplifying the equation. They also show another example with the function of Y in the denominator, which they simplify to cosine and integrate to find the solution.
In this section, we learn how to solve separable equations by recognizing what we're doing with our integrals. These separable equations are not difficult to set up, but the integration techniques may be challenging. We can solve for y easily enough, using an inverse sine function, which results in a general solution with a nice constant term. In solving our example equation, 2/(y(1-x^2)), we group our y's and dy on one side and our x's and dx on the other side, divide by (1-x^2), and integrate by using the cover-up method of partial fractions to find a common denominator. This method is easier and more efficient than attempting a trig substitution.
In this section, the speaker discusses the process of solving for a and b in separable differential equations. The method involves plugging in a value to create factors as zero, and then solving for each variable. Once these variables are found, partial fractions can be used in the integral to solve for y. The speaker goes on to explain some properties of logarithms, such as addition and subtraction resulting in product and quotient, respectively. Finally, to explicitly solve for y, e is taken to both sides, resulting in an equation with absolute value and inverse functions.
In this section, the video covers separable differential equations and techniques for solving them, including using partial fractions and substitutions. The video goes through several examples of separating variables and integrating to find the general solution to the differential equations. One example involves rewriting a power function as a simple linear function and using substitution to find the integral. Another example requires finding a common denominator to simplify the expression and make it easier to reciprocate and solve for the variable. Overall, the video provides a thorough introduction to solving separable differential equations through various techniques and examples.
In this section, the speaker explains how to solve separable differential equations by reciprocating and rearranging the equation. They demonstrate this method through an example problem and then encourage viewers to practice solving a similar problem on their own. The speaker emphasizes the importance of actively practicing and internalizing the material instead of just passively watching the videos, as it will be more beneficial in the long run.
In this section, the presenter introduces the idea of using substitution to solve separable differential equations. They show an example in which they make a substitution to replace a portion of the equation with a new variable, allowing them to integrate the equation more easily. They also discuss alternative ways to deal with constants in the equation and emphasize the importance of looking for separable equations in order to simplify the integration process. The presenter encourages viewers to try working through the example on their own to build their confidence with the techniques.
In this section, the speaker discusses the process for solving an integral by breaking down the terms in the numerator and denominator of a fraction. The goal is to simplify the fraction and make it easier to integrate. The speaker emphasizes that if the fraction only has one term in the denominator, it can be split up to make it simpler to work with. The general solution to the differential equation is found by rearranging the equation and leaving it in its current form, as the process to solve for y would be too complex.
In this section, we learn how to solve separable differential equations. We use the example of y over x equals tangent of y dx over x to demonstrate how to separate the variables and solve for Y. We end up with the expression secant inverse of some constant square root x squared plus one as our general solution.
In this section, the instructor goes over two examples of separable differential equations. In the first example, he explains how to think outside the box when trying to separate variables that don't seem to fit the criteria. In this case, he suggests factoring by grouping and shows how it can create factors that can be separated and solved. In the second example, the problem seems simple at first, but the instructor emphasizes the importance of always separating the variables before integrating.
In this section, the concept of singular solutions for separable differential equations is introduced. It is explained that some solutions may not be part of the general solution and cannot be obtained by normal methods. Singular solutions are those that satisfy the differential equation but are not included in the general solution. The video concludes by encouraging viewers to learn more about the technique and how it can be used with initial value problems.
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