Why Non-Linear Differential Equations are Hard to Solve
In the study of ordinary differential equations, it's well known that most nonlinear equations are pretty much impossible to solve (at least analytically). On the contrary, linear equations are always solvable, and have nice closed form solutions. This is a consequence of the structure of a general solution to the linear equation: it's a sum of two solutions that are easier to find. Solutions to non-linear equations do not always abide by this property. To understand this more deeply, we will briefly study and attempt to solve a small variation of a linear equation and see where the linear method fails. For first order equations, equations involving just the first derivative, a linear equation looks like y' + a(t)y = q(t). Note that there are no weird functions of y in the expression like y^3/2 or sqrt(y), there is just multiplication by functions of t and y and addition of terms, both linear operations. To solve this equation, we would start by looking for two different...