ODEs where every term is of the same degree in the dependent variable and its derivatives.
Definition of Homogeneous ODEs: Homogeneous ODEs are differential equations that can be expressed as a function of only the dependent variable and its derivatives.
Separable ODEs: Separable ODEs can be separated into two parts, one involving only the dependent variable and the other involving only the independent variable.
First-order homogeneous ODEs: Homogeneous first-order ODEs are those equations that can be written in the form dy/dx=f(y/x).
Second-order homogeneous ODEs: Homogeneous second-order ODEs are those equations that can be expressed in the form d²y/dx²=F(y/y').
Linear homogeneous ODEs: Linear homogeneous ODEs can be expressed as a linear combination of the dependent variable, its derivatives, and a constant coefficient.
Integrating factors: Integrating factors are used to solve non-linear first-order ODEs that are not homogeneous in nature.
Eulers ODEs: Euler's ODEs are homogeneous linear differential equations that can be solved using the characteristic equation method.
Laplace transforms: Laplace transforms can be used to solve linear ODEs, including homogeneous ones.
Power series solutions: Power series solutions are used to solve nonlinear ODEs, including homogeneous ones.
Bessel functions: Bessel functions are used to solve second-order homogeneous ODEs with constant coefficients.
Separable ODE: This type of ODE can be written in the form dy/dx=f(x)g(y) and can be solved by separating variables and integrating both sides.
Linear ODE: In a Linear ODE, the dependent variable appears in the equation in a linear form. For example, if we have dy/dx+a(x)y=b(x), where a(x) and b(x) are functions of x, we can solve it using the integrating factor method.
Exact ODE: An Exact ODE has a differential equation that can be expressed as the total derivative of some function. In other words, if we can find a function F(x,y) such that ∂F/∂x + ∂F/∂y = 0, the equation is exact, and the solution can be obtained by integrating along any path that connects two points.
Bernoulli ODE: A Bernoulli ODE is a type of ODE of the form dy/dx+p(x)y=q(x)y^n, where n is not equal to 1. To solve this type of equation, we can use the substitution u=y^(1-n), and the resulting equation becomes linear.
Homogeneous ODE: A Homogeneous ODE is one that is invariant under scaling of the independent and dependent variables. For example, if we have dy/dx=f(y/x), the equation is homogeneous. To solve this type of equation, we can use the substitution y=vx, and the equation becomes separable.
Second-order linear ODE with constant coefficients: A Second-order linear ODE with constant coefficients is a differential equation of the form y''+ay'+by=0, where a and b are constants. The solutions of this equation can be found by assuming a solution of the form y=e^(rt), where r is a constant.
Nonlinear ODE: In a Nonlinear ODE, the dependent variable appears in the equation in a nonlinear form. Examples of nonlinear ODEs include equations that involve trigonometric functions or logarithmic functions.