Mdhiggenz said:Homework Statement
dr/d∅+rtan∅=cos∅
μ(∅)=exp[∫tan∅]
μ∅=exp[-ln[cos∅]=-cos∅
A linear differential equation is an equation that involves a dependent variable, its derivatives, and possibly the independent variable, in a linear manner. This means that the dependent variable and its derivatives appear as part of a linear combination, with no products or powers involved.
To solve a linear differential equation, you need to first identify the type of equation (homogeneous or non-homogeneous) and its order. Then, you can use various methods such as separation of variables, integrating factors, or the method of undetermined coefficients to find the general solution. Boundary conditions can be used to find the particular solution.
A homogeneous linear differential equation has all terms containing the dependent variable and its derivatives, while a non-homogeneous linear differential equation has additional terms that do not involve the dependent variable or its derivatives. This means that the general solution for a homogeneous equation will have a constant of integration, while the general solution for a non-homogeneous equation will have a particular solution added to the general solution of the corresponding homogeneous equation.
Linear differential equations have many applications in physics, engineering, and other fields where a quantity is changing over time. Some examples include modeling the motion of objects, growth and decay processes, and electrical circuits.
Yes, a linear differential equation can have an infinite number of solutions. This is because the general solution contains a constant of integration, which can take on any value. However, a particular solution that satisfies given initial conditions will be unique.