Solutions of a linear, second-order, homogeneous differential equation

Click For Summary
SUMMARY

The discussion centers on proving that a linear, second-order, homogeneous differential equation of the form y'' + P(x)y' + Q(x)y = 0 cannot have three independent solutions using the Wronskian. Participants highlighted the need to construct the Wronskian for three general solutions, y1, y2, and y3, and noted that the Wronskian did not vanish, indicating a flaw in the approach. The conversation emphasizes the importance of understanding the properties of the Wronskian in relation to the linear independence of solutions to differential equations.

PREREQUISITES
  • Understanding of linear, second-order, homogeneous differential equations
  • Familiarity with the Wronskian and its properties
  • Basic knowledge of matrix representation of systems of equations
  • Concept of linear independence in the context of differential equations
NEXT STEPS
  • Study the properties of the Wronskian in detail
  • Learn about the method of solving linear, second-order differential equations
  • Explore examples of linear independence in solutions of differential equations
  • Investigate the implications of the existence of multiple solutions in differential equations
USEFUL FOR

Students and educators in mathematics, particularly those focusing on differential equations, as well as researchers interested in the theoretical aspects of linear independence in solutions.

Holly1990
Messages
1
Reaction score
0
Homework Statement
Show, by means of the Wronskian, that a linear, second-order, homogeneous diffferential equation of the form y''+P(x)y'+Q(x)y=0 cannot have 3 independent solutions.

The attempt at a solution
I tried by constructing a Wronskian of 3 general solutions expecting the wronskian to disappear but it didn't so I'm guessing I've gone about this the wrong way?
 
Physics news on Phys.org
Let y1, y2, and y3 be solutions to the differential equation. That means

y1'' + P(x) y1' + Q(x) y1 = 0
y2'' + P(x) y2' + Q(x) y2 = 0
y3'' + P(x) y3' + Q(x) y3 = 0

Try expressing that system of equations in matrix form.
 

Similar threads

Use Wronskian method in solving the given second order differential equation
  • · Replies 7 ·
Replies
7
Views
2K
Linearly independent functions with identically zero Wronskian
  • · Replies 1 ·
Replies
1
Views
2K
Understanding Wronskian of solutions to homoeneous 2nd order linear DE
  • · Replies 2 ·
Replies
2
Views
2K
Solve (t^2-1)y'' +4ty'+2y=6t, given two particular solutions
  • · Replies 2 ·
Replies
2
Views
1K
Variation of parameter VS Undetermined Coefficients
  • · Replies 7 ·
Replies
7
Views
2K
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
  • · Replies 7 ·
Replies
7
Views
2K
How to Show the Wronskian of Three Distinct Solutions Equals Zero?
  • · Replies 19 ·
Replies
19
Views
4K
Avoid unpleasant integrals in solving IVP
  • · Replies 3 ·
Replies
3
Views
2K
Show that ODE is homogeneous, but I don't think it is
  • · Replies 2 ·
Replies
2
Views
2K
Homogenous solution of a differential equation
  • · Replies 10 ·
Replies
10
Views
2K