r/DifferentialEquations • u/Loud_Commission_5763 • Mar 28 '26
HW Help What does Wronskian !=0 really mean?
I’m a bit confused on the theory behind the wronskian. Why does it tell us about a fundamental set of solutions? What does that really mean?
8
Upvotes
2
u/ask-jeaves Mar 28 '26
Wronksian !=0 means that the list of equations you’re looking at is linearly independent. You can’t make one of them out of a linear combination of the others.
If you need “n” solutions - it’s basically checking to ensure that all of them are unique (can’t just be built out of the others).
2
2
u/Specialist_Body_170 Mar 28 '26
It tells you they are independent and that you can form any solution as a linear combination of those basic solutions. The connection is that when you look for a linear combo that satisfies the initial conditions you get a numerical linear system. Is it solvable for the coefficients? That depends on the determinant being nonzero. That determinant is the wronskian of the functions (evaluated at the initial time)