Mastering Separable Differential Equations

Play an AI-generated podcast conversation about this lesson

Which of the following equations represents the change-of-variables formula in integration?

$dt = \frac{dy},{y(t)}$
$dt = y(t) dy$
$dy = \frac{dt},{y(t)}$
$dy = y(t) dt$ (correct)

What are the objects $h(y) dy$ and $g(t) dt$ called?

Differentiation
Exact equations
Separable equations
Differential forms (correct)

What is the result of integrating both sides of the equation $h(y) dy = g(t) dt$?

$h(y) dy = \int g(t) dt$
$\int h(y) dy = \int g(t) dt$ (correct)
$\int h(y) dy = g(t)$
$h(y) dy = g(t)$

What is the purpose of changing the variable of integration to $y$ in the equation $h(y) dy = g(t) dt$?

To make the equation more meaningful (B) Signup and view all the answers

What is the connection between equations (2.36) and (2.37)?

They are similar when both sides are integrated (D) Signup and view all the answers

What are the special cases like $dy$ and $dt$ called?

Differentials (B) Signup and view all the answers

Which equation represents the solution to the differential equation N = −λN?

$N = Ae−λt$ (A) Signup and view all the answers

What is the half-life of the isotope 32P?

14.3 hours (A) Signup and view all the answers

What is the solution to the differential equation y = ty2?

$y(t) = −1/12 t^2 + C$ (A) Signup and view all the answers

What is the value of λ in equation N = 1000e−0.04861t?

0.04861 (B) Signup and view all the answers

What is the solution to the differential equation N = −λN when A = 0?

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

What step is key in the method of separation of variables?

Going from equation (2.16) to equation (2.17) (B) Signup and view all the answers

Which particle is emitted in the radioactive decay of 238U?

4He (D) Signup and view all the answers

What does the equation N = -λN represent?

The number of nuclei that decay in a given time interval (C) Signup and view all the answers

What is the term used to describe the equation N = -λN?

Exponential equation (A) Signup and view all the answers

What is the purpose of separating the variables in equation (2.4)?

To integrate both sides of the equation (B) Signup and view all the answers

What does the constant λ represent in equation (2.3)?

The rate of decay of radioactive nuclei (C) Signup and view all the answers

What does the equation ln |N| = -λt + C represent?

The number of radioactive nuclei at time t (D) Signup and view all the answers

Which method can be used to solve any separable equation of the form (2.21)?

Integration (A) Signup and view all the answers

According to Newton's law of cooling, the rate of change of an object's temperature (T) is proportional to the difference between its temperature and the ambient temperature (A). Which equation represents this relationship?

dT/dt = k(T - A) (B) Signup and view all the answers

What is the general solution to the differential equation y' = ty^2?

y(t) = -2t^2 + 2C (B) Signup and view all the answers

What is the temperature of the beer as a function of time, according to Newton's law of cooling?

T(t) = A + (T0 - A)e^(-kt) (D) Signup and view all the answers

What is the value of the constant k in the equation T(t) = 70 - 30e^(-0.0405t)?

k = ln(3/2)/10 (D) Signup and view all the answers

What is the exceptional solution to the equation y' = ex/(1+y), given the initial condition y(0) = 1?

y(x) = 1 (C) Signup and view all the answers

Which of the following is an explicit solution to the differential equation in Example 2.25?

$y(x) = -1 + \sqrt{2} + 2e^x$ (C) Signup and view all the answers

What is the interval of existence for the solutions of the differential equation in Example 2.25?

$(-\infty, \infty)$ (C) Signup and view all the answers

What is the form of the implicit equation for the solution of the general differential equation $dy/dt = g(t)/h(y)$?

$H(y) = G(t) + C$ (D) Signup and view all the answers

What is the explicit solution to the differential equation $x' = 2tx(1 + x)$ with initial condition $x(0) = 1$?

$x + \ln(|x|) = t^2 + 1$ (D) Signup and view all the answers

What is the implicit equation for the solution of the differential equation $x' = 2tx(1 + x)$ with initial condition $x(0) = -2$?

$x + \ln(−x) = t^2 + \ln 2 - 2$ (C) Signup and view all the answers

What is the explicit formula for the solution of the differential equation $x' = 2tx(1 + x)$ with initial condition $x(0) = 0$?

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