Linear or Non-Linear Differential Equation
Linear Differential Equations:
- A differential equation is said to be linear if it can be written in the form:
a_n(x) * d^n(y)/dx^n + a_(n-1)(x) * d^(n-1)(y)/dx^(n-1) + ... + a_1(x) * dy/dx + a_0(x) * y = F(x)
- Here, the coefficients a_n(x), a_(n-1)(x), ..., a_1(x), a_0(x) are functions of x, and F(x) is a function of x as well.
Non-Linear Differential Equations:
- A differential equation is considered non-linear if it cannot be written in the above form.

Analysis of the Given Differential Equation
The given differential equation is:
d²y/dx² + x(dy/dx) + sin(y) = 0
This differential equation is non-linear because it contains the term sin(y), which is a non-linear function of y. The presence of this non-linear term makes the entire differential equation non-linear.
Therefore, the given differential equation is a non-linear differential equation.
In conclusion, when analyzing differential equations, it is important to look for non-linear terms like sin(y), cos(y), y^2, etc. These terms indicate that the differential equation is non-linear in nature.