First order, Ordinary Differential Equations.

The video introduces four techniques for solving ordinary differential equations (ODEs): separable equations, homogeneous method, integrating factor, and exact differential equations. It begins with the concept of separable equations, which are solved by isolating variables and integrating. The example given is a first-order linear ODE, which is solved step by step, illustrating the process of converting the equation into a separable form, integrating both sides, and using substitution to find the general solution.

The video explains the homogeneous method, which is applicable when the differential equation is homogeneous. The method involves checking for homogeneity by substituting x with kx and y with ky, and then solving the equation using v-substitution where v equals y/x. The process includes differentiating the substitution, simplifying the equation into a separable form, and integrating to find the solution. An example of a homogeneous differential equation is provided, and the steps to solve it are outlined.

This paragraph delves into solving a specific homogeneous differential equation. It demonstrates the steps of substituting x with kx and y with ky to check for homogeneity, performing v-substitution, and then differentiating and simplifying the equation. The solution process involves separating the variables, integrating both sides, and substituting back to find the explicit form of the solution. The importance of understanding the differentiation of the substitution and the integration techniques is emphasized.

The video outlines the steps for solving ODEs using the homogeneous method, emphasizing the need to check if the function is homogeneous by substituting kx and ky. It details the process of dividing all terms by the highest power of x, performing v-substitution, differentiating, simplifying, and converting the equation into a separable form. The integration of both sides and the substitution of v back into the equation to find the implicit and explicit forms of the solution are also covered.

The integrating factor technique is introduced for solving ODEs that are neither separable nor homogeneous but fit a specific form involving the derivative of y and a function of x. The video explains how to identify the integrating factor, which is the exponential of the integral of the coefficient of y. The process includes multiplying the entire equation by the integrating factor, applying the product rule, and integrating both sides to find the solution. An example is worked through, demonstrating the calculation of the integrating factor and the steps to solve the ODE.

The video discusses the use of integration by parts in solving differential equations, presenting both the tabular method and the formula method. It shows how to select the appropriate functions for u and dv, perform the integration, and apply the method to find the integral of a product of functions. The process is illustrated through an example, highlighting the practicality and efficiency of the tabular method for finding the integral of x times e to the x.

The video explains the concept of exact differential equations and how to identify them by ensuring the given equation is in the standard form involving m(x, y) and n(x, y). It demonstrates the process of checking for exactness by comparing the partial derivatives of m with respect to y and n with respect to x. If they are equal, the equation is exact, and the video outlines the steps for integrating either m or n, differentiating the integrated result, and solving for the function of the other variable.

This paragraph focuses on the integration process for exact ODEs. It describes integrating either m with respect to x or n with respect to y, adding an arbitrary function of the other variable, and differentiating partially to solve for the arbitrary function. The goal is to find an implicit solution by setting the integrated function equal to a constant. The process is illustrated with an example, emphasizing the importance of comparing the differentiated function to the other half of the original equation.

The video concludes with a bonus section on standard integrals, providing a list of useful integrals for exponential, trigonometric, inverse trigonometric, and hyperbolic functions. It also offers a mnemonic technique for remembering the derivatives and anti-derivatives of trigonometric functions using a clockwise and anti-clockwise direction metaphor. The presenter thanks the viewers and invites them to join the next video.