Printable practice problems: Existence & Uniqueness for Linear IVPs

📚 Topic Summary

The Existence and Uniqueness Theorem for Linear Initial Value Problems (IVPs) provides conditions under which we can guarantee that a solution to a linear differential equation exists and is the only solution. For a first-order linear IVP of the form $y' + p(t)y = g(t)$, $y(t_0) = y_0$, if $p(t)$ and $g(t)$ are continuous on an open interval containing $t_0$, then a unique solution exists on that interval. In simpler terms, if the functions in front of $y'$ and $y$, as well as the function on the right side of the equation, are 'nice' (continuous) around your starting point, you're guaranteed a solution, and it's the only one.

Understanding this theorem helps us determine whether our solutions are valid and if we should expect more than one solution (or none at all!). It also guides us in finding the largest possible interval where the solution is valid.

🧠 Part A: Vocabulary

Match the terms with their definitions:

1. Term: Initial Value Problem (IVP)
2. Term: Existence Theorem
3. Term: Uniqueness Theorem
4. Term: Continuous Function
5. Term: Linear Differential Equation

1. Definition: A differential equation together with a specified initial condition.
2. Definition: A theorem that states conditions under which a solution to a differential equation is guaranteed to exist.
3. Definition: A theorem that states conditions under which a solution to a differential equation is the only one.
4. Definition: A function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
5. Definition: A differential equation where the dependent variable and its derivatives appear linearly.

✍️ Part B: Fill in the Blanks

The Existence and Uniqueness Theorem states that for the initial value problem $y' + p(t)y = g(t)$, $y(t_0) = y_0$, if the functions $p(t)$ and $g(t)$ are __________ on an open interval containing $t_0$, then a __________ solution exists on that interval. This means there is one and only one solution satisfying both the __________ and the differential equation.

🤔 Part C: Critical Thinking

Consider the IVP: $y' + \frac{1}{t-2}y = t$, $y(0) = 1$. Does the Existence and Uniqueness Theorem guarantee a unique solution? Explain why or why not.