30 Questions
Which type of differential equations can be solved explicitly?
Linear equations
What are the questions of existence and uniqueness related to?
Initial value problems
What is the normal form of the equation $tx' = x + 3t^2$?
$x' = \frac{1}{t}x + 3t$
What is the general form of the solutions to the initial value problem $tx' = x + 3t^2$, with $x(0) = 1$?
$x(t) = 3t^2 + Ct$
Is there a solution to the initial value problem $tx' = x + 3t^2$, with $x(0) = 1$?
No
Which theorem states that two solutions to the same differential equation that start together stay together?
The uniqueness theorem
What is the purpose of the uniqueness theorem?
To estimate the difference between two solutions to a differential equation
What are the hypotheses of the uniqueness theorem?
All of the above
What is the conclusion of the uniqueness theorem?
Both A and B
How are mathematical theorems different from ordinary logical statements?
Both A and B
According to the uniqueness theorem, what can we conclude about the solution x(t) in relation to the solution y(t) = 1?
x(t) > 1 for all t
According to the text, why is the fact that solution curves cannot meet important?
It limits the space available to any other solution curve
What is the equation being studied in Figure 8?
x(t) = t - x^2
What does the uniqueness theorem assure us about the solution curves shown in Figure 8?
The solution curves never actually meet
Why is the geometric interpretation of solution curves limited in higher order equations and systems of more than one equation?
Curves in higher dimensions do not divide space into separate parts
Which theorem states that given a point (t0, x0) ∈ R, the initial value problem x = f (t, x) and x(t0) = x0 has a solution x(t) defined in an interval containing t0?
Theorem 7.6
What is the interval of existence of the solution to the initial value problem x = 1 + x2 with x(0) = 0?
(-π/2, π/2)
What is the form of a linear equation?
x = a(t)x + g(t)
What is the condition for the existence of solutions for linear equations?
a(t) and g(t) are continuous on the interval b < t < c
What is the only reliable way to discover the interval of existence of a solution?
Find an explicit formula for the solution
Which of the following is an example of a discontinuous function in the context of initial value problems?
$f(t) = 0$
What is the solution to the initial value problem $y' = -2y + 5$, $y(1) = 3e^{-2}$?
$y(t) = 5/2 + (3 - 5e^{2/2})e^{-2t}$
What is the significance of the discontinuity in the solution to the initial value problem in Example 7.10?
The solution is not differentiable everywhere
What does the uniqueness of solutions of initial value problems imply about the physical system being modeled?
The system is deterministic
What is the solution to the initial value problem $x' = x^{1/3}$, $x(0) = 0$?
$x(t) = 0$
According to the uniqueness theorem, what can we conclude if two solution curves of a differential equation meet at a point?
The two solution curves are identical.
What is the main purpose of the existence and uniqueness theorems in differential equations?
To determine if a solution exists.
In Example 7.17, what is the initial condition for the differential equation $tx' = x + 3t^2$?
$x(1) = 2$
In Example 7.18, what is the solution to the differential equation $x' = (x - 1)\cos(xt)$ with the initial condition $x(0) = 1$?
$x(t) = 1$
In Example 7.21, what can we conclude about the solution $x(t)$ of the initial value problem $x' = (x - 1)\cos(xt)$, $x(0) = 2$, if $x(2) = 0$?
$x(t)$ coincides with the solution $y(t) = 1$ for all $t$.
Test your understanding of the theorem involving continuous functions and partial derivatives with this quiz. Explore the relationship between different variables and learn how to apply the theorem to calculate the maximum difference between function values.
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