Theorem involving Continuous Functions and Partial Derivatives Quiz

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Which type of differential equations can be solved explicitly?

Nonlinear equations

Linear equations (correct)

Equations with constant solutions

Equations with undefined coefficients

What are the questions of existence and uniqueness related to?

Initial value problems (correct)

Linear equations

Equations with undefined coefficients

Nonlinear equations

What is the normal form of the equation $tx' = x + 3t^2$?

$x' = \frac{1}{t}x + 3$

$x' = \frac{1}{t}x + 3t$ (correct)

$x' = \frac{1}{t^2}x + 3t$

$x' = \frac{1}{t^2}x + 3$

What is the general form of the solutions to the initial value problem $tx' = x + 3t^2$, with $x(0) = 1$?

$x(t) = 3t^2 + Ct$ Signup and view all the answers

Is there a solution to the initial value problem $tx' = x + 3t^2$, with $x(0) = 1$?

No Signup and view all the answers

Which theorem states that two solutions to the same differential equation that start together stay together?

The uniqueness theorem Signup and view all the answers

What is the purpose of the uniqueness theorem?

To estimate the difference between two solutions to a differential equation Signup and view all the answers

What are the hypotheses of the uniqueness theorem?

All of the above Signup and view all the answers

What is the conclusion of the uniqueness theorem?

Both A and B Signup and view all the answers

How are mathematical theorems different from ordinary logical statements?

Both A and B Signup and view all the answers

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 Signup and view all the answers

According to the text, why is the fact that solution curves cannot meet important?

It limits the space available to any other solution curve Signup and view all the answers

What is the equation being studied in Figure 8?

x(t) = t - x^2 Signup and view all the answers

What does the uniqueness theorem assure us about the solution curves shown in Figure 8?

The solution curves never actually meet Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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) Signup and view all the answers

What is the form of a linear equation?

x = a(t)x + g(t) Signup and view all the answers

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 Signup and view all the answers

What is the only reliable way to discover the interval of existence of a solution?

Find an explicit formula for the solution Signup and view all the answers

Which of the following is an example of a discontinuous function in the context of initial value problems?

$f(t) = 0$ Signup and view all the answers

What is the solution to the initial value problem $y' = -2y + 5$, $y(1) = 3e^{-2}$?

$y(t) = 5/2 + (3 - 5e^{2/2})e^{-2t}$ Signup and view all the answers

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 Signup and view all the answers

What does the uniqueness of solutions of initial value problems imply about the physical system being modeled?

The system is deterministic Signup and view all the answers

What is the solution to the initial value problem $x' = x^{1/3}$, $x(0) = 0$?

$x(t) = 0$ Signup and view all the answers

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. Signup and view all the answers

What is the main purpose of the existence and uniqueness theorems in differential equations?

To determine if a solution exists. Signup and view all the answers

In Example 7.17, what is the initial condition for the differential equation $tx' = x + 3t^2$?

$x(1) = 2$ Signup and view all the answers

In Example 7.18, what is the solution to the differential equation $x' = (x - 1)\cos(xt)$ with the initial condition $x(0) = 1$?

$x(t) = 1$ Signup and view all the answers

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 $t$. Signup and view all the answers