Finding General Solutions: Separation of Variables - AP Calculus AB
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What is $y$ if $\frac{dy}{dx} = xy$ after integrating and solving?
What is $y$ if $\frac{dy}{dx} = xy$ after integrating and solving?
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$y = Ce^{\frac{x^2}{2}}$. Exponential solution from integrating $\frac{dy}{y} = x dx$.
$y = Ce^{\frac{x^2}{2}}$. Exponential solution from integrating $\frac{dy}{y} = x dx$.
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What is the general solution of $\frac{dy}{dx} = \frac{x}{y}$ after separation of variables?
What is the general solution of $\frac{dy}{dx} = \frac{x}{y}$ after separation of variables?
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$y^2 = x^2 + C$. Result from separating $y dy = x dx$ and integrating.
$y^2 = x^2 + C$. Result from separating $y dy = x dx$ and integrating.
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What is the next step after obtaining $y \frac{dy}{dx} = x$?
What is the next step after obtaining $y \frac{dy}{dx} = x$?
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Separate to $y dy = x dx$ and integrate both sides. Variables are now separated for integration.
Separate to $y dy = x dx$ and integrate both sides. Variables are now separated for integration.
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What happens to the constant of integration when finding a particular solution?
What happens to the constant of integration when finding a particular solution?
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It is determined using an initial condition. Initial condition substitutes to find specific $C$ value.
It is determined using an initial condition. Initial condition substitutes to find specific $C$ value.
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What is the result of integrating $e^{-y} dy = x dx$?
What is the result of integrating $e^{-y} dy = x dx$?
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$-e^{-y} = \frac{x^2}{2} + C$. Standard result from integrating separated variables.
$-e^{-y} = \frac{x^2}{2} + C$. Standard result from integrating separated variables.
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What is the solution of $\frac{dy}{dx} = y^2$ using separation of variables?
What is the solution of $\frac{dy}{dx} = y^2$ using separation of variables?
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$-\frac{1}{y} = x + C$. Separate: $y^{-2} dy = dx$, then integrate both sides.
$-\frac{1}{y} = x + C$. Separate: $y^{-2} dy = dx$, then integrate both sides.
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What is the general solution for $\frac{dy}{dx} = 0$?
What is the general solution for $\frac{dy}{dx} = 0$?
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$y = C$, where $C$ is a constant. Zero derivative means $y$ is constant function.
$y = C$, where $C$ is a constant. Zero derivative means $y$ is constant function.
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Separate and integrate: $\frac{dy}{dx} = xy$.
Separate and integrate: $\frac{dy}{dx} = xy$.
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Rewrite as $\frac{1}{y} dy = x dx$ and integrate. Standard separation technique for this product form.
Rewrite as $\frac{1}{y} dy = x dx$ and integrate. Standard separation technique for this product form.
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How do you solve $\frac{dy}{dx} = x e^{y}$ using separation of variables?
How do you solve $\frac{dy}{dx} = x e^{y}$ using separation of variables?
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Rewrite as $e^{-y} dy = x dx$ and integrate. Move exponential to left side before separating.
Rewrite as $e^{-y} dy = x dx$ and integrate. Move exponential to left side before separating.
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Find the error in separation: $y \frac{dy}{dx} = \frac{1}{x}$ rewritten as $y dy = \frac{1}{x} dx$.
Find the error in separation: $y \frac{dy}{dx} = \frac{1}{x}$ rewritten as $y dy = \frac{1}{x} dx$.
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Correct form: $\frac{1}{y} dy = \frac{1}{x} dx$. Division by $y$ was missed in the separation.
Correct form: $\frac{1}{y} dy = \frac{1}{x} dx$. Division by $y$ was missed in the separation.
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What is a key requirement for using separation of variables?
What is a key requirement for using separation of variables?
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The equation must be factored into $N(y) dy = M(x) dx$. Variables must be separable into this product form.
The equation must be factored into $N(y) dy = M(x) dx$. Variables must be separable into this product form.
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How do you separate variables for $\frac{dy}{dx} = \frac{x}{\text{ln}(y)}$?
How do you separate variables for $\frac{dy}{dx} = \frac{x}{\text{ln}(y)}$?
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Rewrite as $\text{ln}(y) dy = x dx$. Move $\ln(y)$ to denominator for proper separation.
Rewrite as $\text{ln}(y) dy = x dx$. Move $\ln(y)$ to denominator for proper separation.
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Separate variables and integrate: $\frac{dy}{dx} = \frac{2x}{y}$.
Separate variables and integrate: $\frac{dy}{dx} = \frac{2x}{y}$.
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$y dy = 2x dx$; integrate to $y^2 = x^2 + C$. Standard separation and integration process.
$y dy = 2x dx$; integrate to $y^2 = x^2 + C$. Standard separation and integration process.
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What is the solution to $\frac{dy}{dx} = y^2 x$?
What is the solution to $\frac{dy}{dx} = y^2 x$?
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$-\frac{1}{y} = \frac{x^2}{2} + C$. Separate: $y^{-2} dy = x dx$, then integrate both sides.
$-\frac{1}{y} = \frac{x^2}{2} + C$. Separate: $y^{-2} dy = x dx$, then integrate both sides.
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What is the role of the constant $C$ in the solution?
What is the role of the constant $C$ in the solution?
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It accounts for the family of solutions. Represents all possible curves in solution family.
It accounts for the family of solutions. Represents all possible curves in solution family.
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What is the general solution of $\frac{dy}{dx} = \frac{1}{x}$?
What is the general solution of $\frac{dy}{dx} = \frac{1}{x}$?
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$y = \text{ln}|x| + C$. Direct integration of $\frac{1}{x}$ with respect to $x$.
$y = \text{ln}|x| + C$. Direct integration of $\frac{1}{x}$ with respect to $x$.
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In separation of variables, what is done after integrating?
In separation of variables, what is done after integrating?
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Solve for $y$ if possible to express $y$ explicitly. Convert implicit form to explicit if possible.
Solve for $y$ if possible to express $y$ explicitly. Convert implicit form to explicit if possible.
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State the form of a differential equation suitable for separation of variables.
State the form of a differential equation suitable for separation of variables.
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$N(y) dy = M(x) dx$ form. Variables separated on opposite sides of equation.
$N(y) dy = M(x) dx$ form. Variables separated on opposite sides of equation.
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Identify the next step: $ \frac{dy}{dx} = \frac{3x}{2y} $ rewritten as $2y dy = 3x dx$.
Identify the next step: $ \frac{dy}{dx} = \frac{3x}{2y} $ rewritten as $2y dy = 3x dx$.
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Integrate both sides: $\frac{y^2}{2} = \frac{3x^2}{2} + C$. Standard integration after proper separation.
Integrate both sides: $\frac{y^2}{2} = \frac{3x^2}{2} + C$. Standard integration after proper separation.
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What is the first step in solving a differential equation using separation of variables?
What is the first step in solving a differential equation using separation of variables?
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Rewrite the equation as $N(y) \frac{dy}{dx} = M(x)$. This isolates variables on separate sides for integration.
Rewrite the equation as $N(y) \frac{dy}{dx} = M(x)$. This isolates variables on separate sides for integration.
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What does it mean if a solution is implicit?
What does it mean if a solution is implicit?
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It is not solved explicitly for $y$ in terms of $x$. $y$ cannot be isolated algebraically from the equation.
It is not solved explicitly for $y$ in terms of $x$. $y$ cannot be isolated algebraically from the equation.
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How is the constant of integration $C$ determined in a particular solution?
How is the constant of integration $C$ determined in a particular solution?
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By using given initial conditions. Substitution yields specific solution from general form.
By using given initial conditions. Substitution yields specific solution from general form.
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What does the equation $dy = M(x) dx$ imply after separation?
What does the equation $dy = M(x) dx$ imply after separation?
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Integrate both sides to find $y = \text{integral of } M(x) dx + C$. Direct integration when variables are separated.
Integrate both sides to find $y = \text{integral of } M(x) dx + C$. Direct integration when variables are separated.
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Find the general solution for $\frac{dy}{dx} = \frac{x}{y}$.
Find the general solution for $\frac{dy}{dx} = \frac{x}{y}$.
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$y^2 = x^2 + C$. Separate variables: $y dy = x dx$, then integrate.
$y^2 = x^2 + C$. Separate variables: $y dy = x dx$, then integrate.
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What do you obtain after integrating both sides of a separated equation?
What do you obtain after integrating both sides of a separated equation?
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An implicit solution, often in terms of $y$ and $x$. Integration produces this general form before solving for $y$.
An implicit solution, often in terms of $y$ and $x$. Integration produces this general form before solving for $y$.
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State the integral of $y \frac{dy}{dx} = x$ after separation.
State the integral of $y \frac{dy}{dx} = x$ after separation.
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$\frac{1}{2}y^2 = \frac{1}{2}x^2 + C$. Direct integration after variable separation.
$\frac{1}{2}y^2 = \frac{1}{2}x^2 + C$. Direct integration after variable separation.
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What form must a differential equation have to use separation of variables?
What form must a differential equation have to use separation of variables?
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The form must be $N(y) \frac{dy}{dx} = M(x)$. Variables must be separable on opposite sides.
The form must be $N(y) \frac{dy}{dx} = M(x)$. Variables must be separable on opposite sides.
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Identify the error: $\frac{dy}{dx} = x^2y$ rewritten as $y \frac{dy}{dx} = x^2$.
Identify the error: $\frac{dy}{dx} = x^2y$ rewritten as $y \frac{dy}{dx} = x^2$.
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Correct: $\frac{1}{y} dy = x^2 dx$. Variables must be properly separated before integrating.
Correct: $\frac{1}{y} dy = x^2 dx$. Variables must be properly separated before integrating.
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What is the general solution of $\frac{dy}{dx} = ky$?
What is the general solution of $\frac{dy}{dx} = ky$?
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$y = Ce^{kx}$, where $C$ is a constant. Standard exponential growth/decay model solution.
$y = Ce^{kx}$, where $C$ is a constant. Standard exponential growth/decay model solution.
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What form does a differential equation take after separation?
What form does a differential equation take after separation?
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$N(y) dy = M(x) dx$. Standard separated form ready for integration.
$N(y) dy = M(x) dx$. Standard separated form ready for integration.
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