30 Questions
Which of the following equations represents the change-of-variables formula in integration?
$dy = y(t) dt$
What are the objects $h(y) dy$ and $g(t) dt$ called?
Differential forms
What is the result of integrating both sides of the equation $h(y) dy = g(t) dt$?
$\int h(y) dy = \int g(t) dt$
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
What is the connection between equations (2.36) and (2.37)?
They are similar when both sides are integrated
What are the special cases like $dy$ and $dt$ called?
Differentials
Which equation represents the solution to the differential equation N = −λN?
$N = Ae−λt$
What is the half-life of the isotope 32P?
14.3 hours
What is the solution to the differential equation y = ty2?
$y(t) = −1/12 t^2 + C$
What is the value of λ in equation N = 1000e−0.04861t?
0.04861
What is the solution to the differential equation N = −λN when A = 0?
$N(t) = 0$
What step is key in the method of separation of variables?
Going from equation (2.16) to equation (2.17)
Which particle is emitted in the radioactive decay of 238U?
4He
What does the equation N = -λN represent?
The number of nuclei that decay in a given time interval
What is the term used to describe the equation N = -λN?
Exponential equation
What is the purpose of separating the variables in equation (2.4)?
To integrate both sides of the equation
What does the constant λ represent in equation (2.3)?
The rate of decay of radioactive nuclei
What does the equation ln |N| = -λt + C represent?
The number of radioactive nuclei at time t
Which method can be used to solve any separable equation of the form (2.21)?
Integration
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)
What is the general solution to the differential equation y' = ty^2?
y(t) = -2t^2 + 2C
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)
What is the value of the constant k in the equation T(t) = 70 - 30e^(-0.0405t)?
k = ln(3/2)/10
What is the exceptional solution to the equation y' = ex/(1+y), given the initial condition y(0) = 1?
y(x) = 1
Which of the following is an explicit solution to the differential equation in Example 2.25?
$y(x) = -1 + \sqrt{2} + 2e^x$
What is the interval of existence for the solutions of the differential equation in Example 2.25?
$(-\infty, \infty)$
What is the form of the implicit equation for the solution of the general differential equation $dy/dt = g(t)/h(y)$?
$H(y) = G(t) + C$
What is the explicit solution to the differential equation $x' = 2tx(1 + x)$ with initial condition $x(0) = 1$?
$x + \ln(|x|) = t^2 + 1$
What is the implicit equation for the solution of the differential equation $x' = 2tx(1 + x)$ with initial condition $x(0) = -2$?
$x + \ln(−x) = t^2 + \ln 2 - 2$
What is the explicit formula for the solution of the differential equation $x' = 2tx(1 + x)$ with initial condition $x(0) = 0$?
$x(t) = 0$
This quiz covers the topic of separable differential equations and provides a step-by-step method to solve them. Learn how to separate the variables, integrate both sides, and solve for the solution. Test your understanding of this important concept in differential equations.
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