Why do the Existence and Uniqueness Theorem and The Principle of Superposition not contradict each other?
I have a question regarding the Existence and Uniqueness Theorem and The Principle of Superposition, which my book (Elementary Differential Equations and Boundary Value Problems) defines in the following ways:
Existence and Uniqueness Theorem
Consider the initial value problem$$y'' + p(t)y' + q(t)y = g(t), \qquad y(t_0) = y_0, \qquad y'(t_0) = y'_0$$where $p$, $q$, and $g$ are continuous on an open interval $I$ that contains the point $t_0$. Then, there is exactly one solution $y = \phi(t)$ of this problem, and the solution exists throughout the interval $I$.
Principle of Superposition
If $y_1$ and $y_2$ are two solutions of the differential equation $y'' + p(t)y' + q(t)y = 0$, then the linear combination $c_1 y_1 + c_2 y_2$ is also a solution for any values of the constants $c_1$ and $c_2$.
From the existence and uniqueness theorem, we know there is only one equation $y = \phi(t)$ that satisfies the equation $y'' + p(t)y' + q(t)y = 0$. From the principle of superposition, we know that $y = c\phi(t)$ is also a solution. How is this possible? Is it because the existence and uniqueness theorem is for particular solutions, where as the principle of superposition is for general solutions?
$\endgroup$2 Answers
$\begingroup$From the existence and uniqueness theorem, we know that given $y_0$ and $y'_0$, there is only one solution $y = \phi(t)$ (depending on $y_0$ and $y'_0$) of the equation $y'' + p(t)y' + q(t)y = 0$ satisfying $y(t_0) = y_0$ and $y'(t_0) = y'_0$.
But when you make linear combinations of solutions you change their values $y_0$ and $y'_0$.
$\endgroup$ $\begingroup$Existence and Uniqueness Theorem doesn't say that there is only one function $y = \phi(t)$ satisfying the equation $y'' + p(t)y' + q(t)y = 0$. In fact, the number of solutions for this equation is infinite.
Existence and Uniqueness Theorem says that there is only one function $y = \phi(t)$ satisfying simultaneously the equation $y'' + p(t)y' + q(t)y = 0$ and the initial conditions $y(t_0) = y_0$, $y'(t_0) = y'_0$.
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