Solve 3rd Order DE: How Many Linearly Independent Solutions?

[bigdirtycarl]

This is a quick one: Is there a way to tell how many linearly independent solutions there are to a differential equation? Like by its order? I have to solve a third order differential equation, and I am wondering how many possible linearly independent solutions there are. Three perhaps? Thanks!

 

[lurflurf]

In general no, though if the equation and initial conditions meet some conditions the numbers of constants will equal the order.

 

[HallsofIvy]

In order to talk about "linearly independent" solutions, you must be talking about linear differential equations.

The set of all solutions to a linear, homogeneous, differential equation, of order n, forms a vector space of dimension n so, yes, there are 3 "linearly independent" solutions to a third degree linear differential equation.

To prove this, consider the "fundamental" solutions, at some x= a. For an n degree equation, the fundamental solutions are solutions to the differential equation satisfying:
for $y^(m)_n(a)= \delta_{mn}$. That is, the nth derivative of $y_n$ at x= a is 1 and all other derivatives are 0 there. (The "0" derivative is the value of the function.)

It is easy to show that those are "linearly independent" functions. Further if y(x) is any solution to the differential equation and [math]Y_n[/math] is the nth derivative of y at x= a, then [math]y(x)= \sum_{i= 0}^n Y_i y_i(x)[/math] so they also span the space.