Modeling Situations with Differential Equations - AP Calculus AB
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Which method is used to solve $y' = yx$?
Which method is used to solve $y' = yx$?
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Separation of variables. Variables can be separated: $\frac{dy}{y} = x dx$.
Separation of variables. Variables can be separated: $\frac{dy}{y} = x dx$.
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What is a homogeneous differential equation?
What is a homogeneous differential equation?
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All terms are a function of the dependent variable and its derivatives. No external forcing term, only function and derivatives.
All terms are a function of the dependent variable and its derivatives. No external forcing term, only function and derivatives.
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What is the solution to $\frac{dy}{dx} = 3x^2$ with $y(0)=4$?
What is the solution to $\frac{dy}{dx} = 3x^2$ with $y(0)=4$?
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$y = x^3 + 4$. Integrate $3x^2$ and apply the initial condition.
$y = x^3 + 4$. Integrate $3x^2$ and apply the initial condition.
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Identify the independent variable in $dy/dt = -2y$.
Identify the independent variable in $dy/dt = -2y$.
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$t$. The variable in the denominator of the derivative.
$t$. The variable in the denominator of the derivative.
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What does it mean for a DE to be linear?
What does it mean for a DE to be linear?
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Dependent variable and its derivatives are linear. No powers or products of $y$ or its derivatives.
Dependent variable and its derivatives are linear. No powers or products of $y$ or its derivatives.
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What is an initial condition in differential equations?
What is an initial condition in differential equations?
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A value that specifies the solution of a DE at a point. Determines a unique solution from the general solution.
A value that specifies the solution of a DE at a point. Determines a unique solution from the general solution.
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What method is used to solve $y' = xy$?
What method is used to solve $y' = xy$?
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Separation of variables. Can separate variables: $\frac{dy}{y} = x dx$.
Separation of variables. Can separate variables: $\frac{dy}{y} = x dx$.
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What is a differential equation?
What is a differential equation?
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An equation involving derivatives of a function. Relates a function to its rate of change.
An equation involving derivatives of a function. Relates a function to its rate of change.
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What is a first-order differential equation?
What is a first-order differential equation?
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An equation involving the first derivative of a function. Only involves $y'$, no higher derivatives.
An equation involving the first derivative of a function. Only involves $y'$, no higher derivatives.
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Find the particular solution of $y' = 2x$ given $y(1) = 3$.
Find the particular solution of $y' = 2x$ given $y(1) = 3$.
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$y = x^2 + 2$. Integrate $2x$ to get $x^2 + C$, then use initial condition.
$y = x^2 + 2$. Integrate $2x$ to get $x^2 + C$, then use initial condition.
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Which function satisfies $y' = 0$?
Which function satisfies $y' = 0$?
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Constant functions. Zero derivative means no change, so $y$ is constant.
Constant functions. Zero derivative means no change, so $y$ is constant.
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Identify the dependent variable in $dy/dx = x^2y$.
Identify the dependent variable in $dy/dx = x^2y$.
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$y$. The variable being differentiated ($y$ depends on $x$).
$y$. The variable being differentiated ($y$ depends on $x$).
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State the general solution for $y' = \frac{1}{2}y$.
State the general solution for $y' = \frac{1}{2}y$.
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$y = Ce^{\frac{1}{2}x}$. Exponential solution with growth rate $k = \frac{1}{2}$.
$y = Ce^{\frac{1}{2}x}$. Exponential solution with growth rate $k = \frac{1}{2}$.
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Find the integrating factor for $y' + 2y = 3x$.
Find the integrating factor for $y' + 2y = 3x$.
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$e^{2x}$. For $y' + py = q$, integrating factor is $e^{\int p dx} = e^{2x}$.
$e^{2x}$. For $y' + py = q$, integrating factor is $e^{\int p dx} = e^{2x}$.
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What technique solves $y' + p(x)y = q(x)$?
What technique solves $y' + p(x)y = q(x)$?
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Integrating factor. Standard method for linear first-order equations.
Integrating factor. Standard method for linear first-order equations.
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What is the solution to $y' + y = 0$ with $y(0) = 3$?
What is the solution to $y' + y = 0$ with $y(0) = 3$?
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$y = 3e^{-x}$. Exponential decay with initial condition applied.
$y = 3e^{-x}$. Exponential decay with initial condition applied.
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Identify the particular solution of $y' = 2y$ given $y(1) = 3$.
Identify the particular solution of $y' = 2y$ given $y(1) = 3$.
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$y = 3e^{2x - 2}$. Apply exponential solution and initial condition at $x = 1$.
$y = 3e^{2x - 2}$. Apply exponential solution and initial condition at $x = 1$.
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Identify the order of the differential equation $y'' + 3y' - 5y = 0$.
Identify the order of the differential equation $y'' + 3y' - 5y = 0$.
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Second order. The highest derivative is $y''$ (second derivative).
Second order. The highest derivative is $y''$ (second derivative).
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What is the general solution of $y' = ky$?
What is the general solution of $y' = ky$?
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$y = Ce^{kx}$. Exponential growth/decay solution form.
$y = Ce^{kx}$. Exponential growth/decay solution form.
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What is the order of the DE $x^2y'' + xy' + y = 0$?
What is the order of the DE $x^2y'' + xy' + y = 0$?
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Second order. Highest derivative present is the second derivative $y''$.
Second order. Highest derivative present is the second derivative $y''$.
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State the form of a separable differential equation.
State the form of a separable differential equation.
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$\frac{dy}{dx} = g(x)h(y)$. Variables can be separated to each side of the equation.
$\frac{dy}{dx} = g(x)h(y)$. Variables can be separated to each side of the equation.
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What is the general solution of $y' = 0$?
What is the general solution of $y' = 0$?
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Constant function. No change means the function remains constant.
Constant function. No change means the function remains constant.
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What is a solution to $y' = y$ with $y(0) = 2$?
What is a solution to $y' = y$ with $y(0) = 2$?
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$y = 2e^x$. Exponential solution $y = Ce^x$ with $C = 2$ from condition.
$y = 2e^x$. Exponential solution $y = Ce^x$ with $C = 2$ from condition.
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What is a particular solution for $y' = xy$ given $y(1) = 4$?
What is a particular solution for $y' = xy$ given $y(1) = 4$?
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$y = 4e^{\frac{x^2 - 1}{2}}$. Separate variables and integrate, then apply condition.
$y = 4e^{\frac{x^2 - 1}{2}}$. Separate variables and integrate, then apply condition.
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What does it mean for a DE to be separable?
What does it mean for a DE to be separable?
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It can be written as $g(y)dy = f(x)dx$. Variables can be moved to opposite sides for integration.
It can be written as $g(y)dy = f(x)dx$. Variables can be moved to opposite sides for integration.
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What is a particular solution of a differential equation?
What is a particular solution of a differential equation?
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A solution satisfying the initial conditions. General solution with constants determined by conditions.
A solution satisfying the initial conditions. General solution with constants determined by conditions.
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What is the integrating factor for $y' + 3y = 0$?
What is the integrating factor for $y' + 3y = 0$?
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$e^{3x}$. For homogeneous equation, integrating factor is $e^{3x}$.
$e^{3x}$. For homogeneous equation, integrating factor is $e^{3x}$.
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What is the solution to $dy/dx = 3x^2$ with $y(0) = 1$?
What is the solution to $dy/dx = 3x^2$ with $y(0) = 1$?
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$y = x^3 + 1$. Integrate and apply the given initial condition.
$y = x^3 + 1$. Integrate and apply the given initial condition.
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What is the form of a linear first-order DE?
What is the form of a linear first-order DE?
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$y' + p(x)y = q(x)$. Standard form for first-order linear differential equations.
$y' + p(x)y = q(x)$. Standard form for first-order linear differential equations.
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What is the solution to $y' = 3y$ with $y(0) = 5$?
What is the solution to $y' = 3y$ with $y(0) = 5$?
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$y = 5e^{3x}$. Exponential solution with growth constant 3 and initial value 5.
$y = 5e^{3x}$. Exponential solution with growth constant 3 and initial value 5.
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