A first-order ordinary differential equation (ODE I) is a mathematical equation that describes the relationship between a function and its derivative. There are several types of first-order ODEs, including separable, homogeneous, linear, exact, Bernoulli, Riccati, and nonlinear ODEs. Each type has its own characteristic form and requires different methods for solving.
Types of ODE(I)
Separable ODEs: A separable ODE is one that can be written in the form \(y\prime(x) = f(x)g(y)\), where \(f(x)\) and \(g(y)\) are functions of \(x\) and \(y\), respectively. To solve a separable ODE, we can separate the variables and integrate both sides with respect to \(x\) and \(y\), respectively.
Homogeneous ODEs: A homogeneous ODE is one that can be written in the form \(y\prime(x) = f(\frac{y}{x})\), where \(f\) is a function of \(\frac{y}{x}\). To solve a homogeneous ODE, we can use the substitution \(u = \frac{y}{x}\) to transform the equation into a separable ODE.
Linear ODEs: A linear ODE is one that can be written in the form \(y\prime(x) + p(x)y(x) = q(x)\), where \(p(x)\) and \(q(x)\) are functions of \(x\). To solve a linear ODE, we can use an integrating factor, which is a function that makes the left-hand side of the equation equal to the derivative of a product. Then, we can integrate both sides to solve for \(y(x)\).
Exact ODEs: An exact ODE is one that can be written in the form \(M(x,y)dx + N(x,y)dy = 0\), where \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). To solve an exact ODE, we can find a function \(\phi(x,y)\) such that \(\frac{\partial \phi}{\partial x} = M(x,y)\) and \(\frac{\partial \phi}{\partial y} = N(x,y)\). Then, the general solution is given by \(\phi(x,y) = C\), where \(C\) is a constant.
Bernoulli ODEs: A Bernoulli ODE is one that can be written in the form \(y\prime(x) + p(x)y(x) = q(x)y(x)^n\), where \(n\) is a constant. To solve a Bernoulli ODE, we can use the substitution \(u = y^{(1-n)}\) to transform the equation into a linear ODE.
Riccati ODEs: A Riccati ODE is one that can be written in the form \(y\prime(x) = p(x)y(x)^2 + q(x)y(x) + r(x)\), where \(p(x)\), \(q(x)\), and \(r(x)\) are functions of \(x\). To solve a Riccati ODE, we can use a substitution \(u = \frac{y\prime(x)}{y(x)}\) to transform the equation into a linear ODE.
Nonlinear ODEs: A nonlinear ODE is one that cannot be written in any of the above forms. These can be more difficult to solve and often require numerical methods to approximate the solution. Some common numerical methods include Euler's method, the Runge-Kutta method, and the shooting method.
I also would like to mention the special case of first-order ordinary differential equations. These equations can be solved by predicting the form of the solution.
Some of these first-order ordinary differential equations are described in the video: