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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?

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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 $y'_0$.

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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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  • $\begingroup$ Thanks! This is what I suspected, but I wanted to confirm my suspicions. $\endgroup$ – Birdman2246 Feb 2 '16 at 1:47
  • $\begingroup$ @Birdman2246 You are welcome! In short: they talk about different things. Principle of Superposition talks about homogeneous equations but Existence and Uniqueness Theorem talks about initial value problems. $\endgroup$ – Pedro Feb 2 '16 at 1:51

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