You CAN Ace Differential Equations
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Single Variable Calculus |
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Acceleration Vector |
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Cylindrical Coordinates |
Lagrange Multipliers |
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Differential Equations |
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For an equation of the type \(y' = p(x)y + q(x)y^n\), called a Bernoulli Equation, we can use the special substitution \(v = y^{1-n}\), which will turn the equation into a linear equation.
Note: This technique uses integrating factors in order to solve the resulting linear equation.
Here are several good videos explaining the theory of how and why this substitution works.
video by PatrickJMT
video by Engineer4Free
video by MIT OCW
Instructions - - Unless otherwise instructed, solve these Bernoulli Equations. Give your answers in exact form.
\(yâ€™=(A\cos t+B)y-y^3\)
Problem Statement |
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Solve the Bernoulli Equation \(yâ€™=(A\cos t+B)y-y^3\).
Solution |
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video by PatrickJMT
close solution |
\(t^2y'+2ty-y^3=0\)
Problem Statement |
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Solve the Bernoulli Equation \(t^2y'+2ty-y^3=0\).
Solution |
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We found two solutions to this problem presented by two different people. If the first one doesn't help you, try the second one.
video by PatrickJMT
video by Engineer4Free
close solution |
\(y'+xy=xy^2\)
Problem Statement |
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Solve the Bernoulli Equation \(y'+xy=xy^2\).
Solution |
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video by MIP4U
close solution |
\(\displaystyle{ xy'+y=\frac{1}{y^2} }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ xy'+y=\frac{1}{y^2} }\).
Solution |
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video by MIP4U
close solution |
\(\displaystyle{ \frac{dy}{dx} + y = xy^4 }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dx} + y = xy^4 }\).
Solution |
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video by Houston Math Prep
close solution |
\(\displaystyle{ \frac{dy}{dx} = \sqrt{y} - y; y(0) = 9 }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dx} = \sqrt{y} - y; y(0) = 9 }\).
Solution |
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video by blackpenredpen
close solution |
\(\displaystyle{ \frac{dy}{dx} - \frac{1}{x}y = xy^2 }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dx} - \frac{1}{x}y = xy^2 }\).
Solution |
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video by Houston Math Prep
close solution |
\(\displaystyle{ y' + \frac{2}{x}y = -x^9y^5; y(1) = 1 }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ y' + \frac{2}{x}y = -x^9y^5; y(1) = 1 }\).
Solution |
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video by MIP4U
close solution |
\( y' = -y(1 + xy^2) \)
Problem Statement |
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Solve the Bernoulli Equation \( y' = -y(1 + xy^2) \).
Solution |
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video by blackpenredpen
close solution |
\(\displaystyle{ y' + \frac{4}{x} y = x^3y^2 }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ y' + \frac{4}{x} y = x^3y^2 }\).
Solution |
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This problem was solved by three different people. We include all three videos here in case one helps you more than the other.
video by The Lazy Engineer
video by MIP4U
close solution |
\( y' + 2xy = xy^3 \)
Problem Statement |
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Solve the Bernoulli Equation \( y' + 2xy = xy^3 \).
Solution |
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video by Houston Math Prep
close solution |
\(\displaystyle{ y' + \frac{1}{t}y = -ty^3 }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ y' + \frac{1}{t}y = -ty^3 }\).
Solution |
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video by Engineer4Free
close solution |
\(\displaystyle{ \frac{dy}{dt} = 2y + y^5; y(0) = 1 }\)
Problem Statement |
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Solve the Bernoulli Equation \(\displaystyle{ \frac{dy}{dt} = 2y + y^5; y(0) = 1 }\).
Solution |
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video by MIP4U
close solution |