Uniqueness and existence condition of solution of differential equation
Answers
Recall that for a first order linear differential equation
y' + p(t)y = g(t) y(t0) = y0
if p(t) and g(t) are continuous on [a,b], then there exists a unique solution on the interval [a,b].
We can ask the same questions of second order linear differential equations. We need to first make a few comments. The first is that for a second order differential equation, it is not enough to state the initial position. We must also have the initial velocity. One way of convincing yourself, is that since we need to reverse two derivatives, two constants of integration will be introduced, hence two pieces of information must be found to determine the constants.
A second comment is that of notation. Let
y'' + p(t)y' + q(t)y = g(t)
be a second order linear differential equation. Then we call the operator
L(y) = y'' + p(t)y' + q(t)y
the corresponding linear operator. Thus we want to find solutions to the equation
L(y) = g(t) y(t0) = y0 y'(t0) = y'0
We will state the following theorem without proof. The proof is well above the level of this course.
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