First-order ODEs

Integrals: Understanding integrals is critical to solving differential equations. You'll need to be comfortable with integrating functions and differentiating polynomials.

Linear Equations: Linear equations are a type of first-order ODE that can be solved relatively easily. It's essential to learn how to identify them and solve them.

Separable Equations: Separable equations are another type of first-order ODE that can be broken down and solved more easily. Learn how to identify them and solve them using integration.

Homogeneous Equations: Ho unique equations are ODEs that feature terms that are proportional to each other. These equations require special methods to solve.

Exact Equations: Exact equations are first-order ODEs that have a special property that makes them easy to solve.

Bernoulli Equations: Bernoulli equations are nonlinear ODEs that can be transformed into linear equations and solved using the technique for linear equations.

Solving ODEs Numerically: You'll need to learn how to use numerical methods to solve differential equations when they can't be solved analytically.

Laplace Transforms: Laplace transforms are a powerful tool for solving ODEs. You'll need to learn how to use them to simplify differential equations.

Method of Undetermined Coefficients: The method of undetermined coefficients is an effective tool for solving linear ODEs featuring nonhomogeneous terms.

Variation of Parameters: Variation of parameters is another method for solving nonhomogeneous linear ODEs.

Higher-order ODEs: Once you've mastered first-order ODEs, you'll need to move on to higher-order ODEs. Learn how to convert them to equivalent first-order equations for easier solving.

Applications: ODEs are used in many areas of science and engineering, such as modeling population growth, chemical reactions, and electrical circuits. Understanding these applications is essential to becoming proficient in solving ODEs.

Separable: An equation of the form y'(x) = g(x)h(y(x)), where g and h are known functions. The solution can be found by separating the variables and integrating.

Linear: An equation of the form y'(x) + p(x)y(x) = q(x), where p and q are known functions. The solution involves finding an integrating factor and then integrating.

Homogeneous: An equation of the form y'(x) = f(y/x), where f is a known function. The solution can be found by making the substitution y(x) = ux and solving for u using separation of variables.

Bernoulli: An equation of the form y'(x) + p(x)y(x) = q(x)y(x)^n, where p, q, and n are known constants. The solution can be found by making the substitution z(x) = y(x)^(1-n) and solving a linear equation.

Exact: An equation of the form M(x,y)dx + N(x,y)dy = 0, where M and N have continuous first-order partial derivatives with respect to y and x, respectively, and M_y = N_x. The solution can be found by integrating a function that is a total differential of the solution.

Bernoulli exact: A mixture of the Bernoulli and exact equations.

Riccati: An equation of the form y'(x) = p(x)y(x) + q(x)y(x)^2 + r(x), where p, q, and r are known functions. The solution involves finding a particular solution and then making a substitution which reduces the equation to a linear equation.

Clairaut: An equation of the form y(x) = xy'(x) + f(y'), where f is a known function. The solution can be found using a standard equation for finding general solutions to first-order differential equations.