Modeling Situations with Differential Equations - AP Calculus BC
Card 1 of 30
Define a linear differential equation.
Define a linear differential equation.
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An equation where the dependent variable and derivatives appear linearly. Dependent variable appears to first power only.
An equation where the dependent variable and derivatives appear linearly. Dependent variable appears to first power only.
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Write the differential equation for exponential growth.
Write the differential equation for exponential growth.
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$\frac{dy}{dt} = ky$. Rate proportional to current amount.
$\frac{dy}{dt} = ky$. Rate proportional to current amount.
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What is the integrating factor for $dy/dx + 2y = e^x$?
What is the integrating factor for $dy/dx + 2y = e^x$?
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$e^{2x}$. Exponential of integral of coefficient: $e^{\int 2 dx}$.
$e^{2x}$. Exponential of integral of coefficient: $e^{\int 2 dx}$.
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What is a particular solution?
What is a particular solution?
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A solution satisfying both the differential equation and initial conditions. Specific solution meeting initial conditions.
A solution satisfying both the differential equation and initial conditions. Specific solution meeting initial conditions.
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Identify the dependent variable in $dz/dt = z^2 + t^2$.
Identify the dependent variable in $dz/dt = z^2 + t^2$.
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The dependent variable is $z$. Function being differentiated ($z$ depends on $t$).
The dependent variable is $z$. Function being differentiated ($z$ depends on $t$).
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Identify the dependent variable in $dy/dx = x^2 + y^2$.
Identify the dependent variable in $dy/dx = x^2 + y^2$.
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The dependent variable is $y$. Variable being differentiated ($y$ depends on $x$).
The dependent variable is $y$. Variable being differentiated ($y$ depends on $x$).
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What is the integrating factor for $y' + p(x)y = g(x)$?
What is the integrating factor for $y' + p(x)y = g(x)$?
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$e^{\int p(x) \text{d}x}$. Multiplier that makes equation exact.
$e^{\int p(x) \text{d}x}$. Multiplier that makes equation exact.
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State a real-world example of a first-order differential equation.
State a real-world example of a first-order differential equation.
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Newton's Law of Cooling. Temperature change proportional to difference.
Newton's Law of Cooling. Temperature change proportional to difference.
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What is the general solution to $y'' + 4y = 0$?
What is the general solution to $y'' + 4y = 0$?
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$y = C_1 \text{cos}(2x) + C_2 \text{sin}(2x)$. Characteristic equation $r^2 + 4 = 0$ gives $r = ±2i$.
$y = C_1 \text{cos}(2x) + C_2 \text{sin}(2x)$. Characteristic equation $r^2 + 4 = 0$ gives $r = ±2i$.
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What method is used to solve $y' = ky$?
What method is used to solve $y' = ky$?
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Separation of variables. Rearrange to separate $y$ and $x$ terms.
Separation of variables. Rearrange to separate $y$ and $x$ terms.
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What is the general solution to $y'' + 9y = 0$?
What is the general solution to $y'' + 9y = 0$?
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$y = C_1 \text{cos}(3x) + C_2 \text{sin}(3x)$. Characteristic equation $r^2 + 9 = 0$ gives $r = ±3i$.
$y = C_1 \text{cos}(3x) + C_2 \text{sin}(3x)$. Characteristic equation $r^2 + 9 = 0$ gives $r = ±3i$.
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What does it mean for a function to be a solution to a differential equation?
What does it mean for a function to be a solution to a differential equation?
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Substituting it into the equation yields a true statement. The function satisfies the equation identically.
Substituting it into the equation yields a true statement. The function satisfies the equation identically.
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What is the characteristic equation of $y'' + 2y' + 5y = 0$?
What is the characteristic equation of $y'' + 2y' + 5y = 0$?
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$r^2 + 2r + 5 = 0$. Substitute $y'' = r^2$, $y' = r$ into equation.
$r^2 + 2r + 5 = 0$. Substitute $y'' = r^2$, $y' = r$ into equation.
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Define a non-homogeneous differential equation.
Define a non-homogeneous differential equation.
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An equation with non-zero terms not involving the dependent variable. Contains terms independent of dependent variable.
An equation with non-zero terms not involving the dependent variable. Contains terms independent of dependent variable.
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Identify the homogeneous part of $y'' + 5y' + 6y = e^x$.
Identify the homogeneous part of $y'' + 5y' + 6y = e^x$.
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$y'' + 5y' + 6y$. Left side when right side equals zero.
$y'' + 5y' + 6y$. Left side when right side equals zero.
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Solve $y' = \frac{1}{x}$ for $y$.
Solve $y' = \frac{1}{x}$ for $y$.
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$y = \text{ln}|x| + C$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$.
$y = \text{ln}|x| + C$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$.
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What function solves $y' - 2y = 0$?
What function solves $y' - 2y = 0$?
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$y = Ce^{2x}$. Exponential solution with coefficient $k = 2$.
$y = Ce^{2x}$. Exponential solution with coefficient $k = 2$.
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What is the solution to $y'' - y = 0$?
What is the solution to $y'' - y = 0$?
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$y = C_1e^x + C_2e^{-x}$. Characteristic equation $r^2 - 1 = 0$ gives $r = ±1$.
$y = C_1e^x + C_2e^{-x}$. Characteristic equation $r^2 - 1 = 0$ gives $r = ±1$.
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Identify the linear part of $y'' + 4y' + 4y = 5x$.
Identify the linear part of $y'' + 4y' + 4y = 5x$.
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$y'' + 4y' + 4y$. Terms involving $y$ and its derivatives.
$y'' + 4y' + 4y$. Terms involving $y$ and its derivatives.
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What is the solution for $dy/dx = y$ using separation of variables?
What is the solution for $dy/dx = y$ using separation of variables?
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$y = Ce^{x}$. Separate variables: $\frac{dy}{y} = dx$, integrate.
$y = Ce^{x}$. Separate variables: $\frac{dy}{y} = dx$, integrate.
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What is the characteristic equation of $y'' + 4y' + 4y = 0$?
What is the characteristic equation of $y'' + 4y' + 4y = 0$?
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$r^2 + 4r + 4 = 0$. Replace $y''$ with $r^2$, $y'$ with $r$.
$r^2 + 4r + 4 = 0$. Replace $y''$ with $r^2$, $y'$ with $r$.
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State the general solution form for $y' = ky$.
State the general solution form for $y' = ky$.
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$y = Ce^{kt}$, where $C$ is a constant. Exponential growth/decay model solution.
$y = Ce^{kt}$, where $C$ is a constant. Exponential growth/decay model solution.
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Define a separable differential equation.
Define a separable differential equation.
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An equation where variables can be separated on opposite sides. Variables can be moved to opposite sides.
An equation where variables can be separated on opposite sides. Variables can be moved to opposite sides.
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What is a homogeneous differential equation?
What is a homogeneous differential equation?
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An equation where $0$ is a solution when all terms set to $0$. All terms involve the dependent variable.
An equation where $0$ is a solution when all terms set to $0$. All terms involve the dependent variable.
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Identify the type of differential equation: $y'' + y = \tan(x)$.
Identify the type of differential equation: $y'' + y = \tan(x)$.
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Second-order non-homogeneous differential equation. Second-order with non-zero right side.
Second-order non-homogeneous differential equation. Second-order with non-zero right side.
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Solve for $y$ in the differential equation $\frac{dy}{dx} = 0$.
Solve for $y$ in the differential equation $\frac{dy}{dx} = 0$.
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$y = C$, where $C$ is a constant. Zero derivative means constant function.
$y = C$, where $C$ is a constant. Zero derivative means constant function.
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What is the order of $d^2y/dx^2 + 3y' + 4y = x^2$?
What is the order of $d^2y/dx^2 + 3y' + 4y = x^2$?
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Order is 2. Highest derivative is second order.
Order is 2. Highest derivative is second order.
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Identify the independent variable in $dy/dx = x^2 + y^2$.
Identify the independent variable in $dy/dx = x^2 + y^2$.
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The independent variable is $x$. Variable with respect to which we differentiate.
The independent variable is $x$. Variable with respect to which we differentiate.
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What type of differential equation is $y' + p(x)y = g(x)$?
What type of differential equation is $y' + p(x)y = g(x)$?
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First-order linear differential equation. Standard form with $y$ and $y'$ terms only.
First-order linear differential equation. Standard form with $y$ and $y'$ terms only.
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Find the derivative of $y = Ce^{kt}$.
Find the derivative of $y = Ce^{kt}$.
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$y' = kCe^{kt}$. Apply chain rule: $\frac{d}{dt}(Ce^{kt}) = kCe^{kt}$.
$y' = kCe^{kt}$. Apply chain rule: $\frac{d}{dt}(Ce^{kt}) = kCe^{kt}$.
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