A linear differential equation in strange form is a type of differential equation in which the dependent variable and its derivatives are multiplied by some functions that are not necessarily continuous or differentiable. The equation takes the form of y' = f(x)y + g(x)y2, where y is the dependent variable, y' is its derivative, and f(x) and g(x) are functions of the independent variable x.
The main difference between a linear differential equation in strange form and a standard linear differential equation is the presence of non-continuous or non-differentiable functions in the equation. In a standard linear differential equation, the functions are usually continuous and differentiable.
Some examples of linear differential equations in strange form are: y' = x2y + cos(x)y2, y' = exy + ln(x)y2, and y' = sin(x)y + sqrt(x)y2.
To solve a linear differential equation in strange form, you can use the method of separation of variables or the method of integrating factors. Both methods involve manipulating the equation to separate the variables and then integrating to find the solution.
Linear differential equations in strange form have applications in various fields such as physics, engineering, and economics. They can be used to model real-life situations involving changing quantities, such as population growth or chemical reactions. They also have applications in control systems, where they are used to design controllers for systems with nonlinear dynamics.