hsong9 said:Homework Statement
Can y = sin(t^2) be a solution on an interval containing t = 0 of an equation y'' + p(t)y' + q(t)y = 0 with continuous coefficients?
Homework Equations
The Attempt at a Solution
y = sin(t^2)
y' = 2tcos(t^2)
y'' = 2cos(t^2) - 4t^2sin(t^2)
2cos(t^2) - 4t^2sin(t^2) + p(t)(2tcos(t^2)) + q(t)sin(t^2) = 0
when t=0, above equation is 2. That is, there does not exist the solution. so y can not be a solution on I containing t=0.
A fundamental solution of a linear homogeneous equation is a set of linearly independent solutions that can be used to express any other solution of the equation. In other words, it is a set of solutions that form a basis for the solution space.
A set of solutions is a fundamental solution if they satisfy two conditions: they are linearly independent and they span the solution space. This means that none of the solutions can be written as a linear combination of the others, and any other solution of the equation can be written as a linear combination of the fundamental solutions.
Yes, a linear homogeneous equation can have an infinite number of fundamental solutions. In fact, any linear combination of fundamental solutions is also a fundamental solution.
The general solution of a linear homogeneous equation is a linear combination of the fundamental solutions. This means that by finding the fundamental solutions, we can find the general solution of the equation.
Fundamental solutions are important because they provide a way to find the general solution of a linear homogeneous equation. They also help us understand the structure of the solution space and the behavior of the solutions of the equation.