You CAN Ace Differential Equations

 integrating factors - linear

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > differential equations > bernoulli equation

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.

### PatrickJMT - Bernoulli Equation for Differential Equations , Part 1 [10mins-25secs]

video by PatrickJMT

### Engineer4Free - How to solve Bernoulli differential equations [5mins-29secs]

video by Engineer4Free

video by MIT OCW

### Practice

Instructions - - Unless otherwise instructed, solve these Bernoulli Equations. Give your answers in exact form.

$$y’=(A\cos t+B)y-y^3$$

Problem Statement

Solve the Bernoulli Equation $$y’=(A\cos t+B)y-y^3$$.

Solution

### 2417 solution video

video by PatrickJMT

$$t^2y'+2ty-y^3=0$$

Problem Statement

Solve the Bernoulli Equation $$t^2y'+2ty-y^3=0$$.

Solution

We found two solutions to this problem presented by two different people. If the first one doesn't help you, try the second one.

### 2418 solution video

video by PatrickJMT

### 2418 solution video

video by Engineer4Free

$$y'+xy=xy^2$$

Problem Statement

Solve the Bernoulli Equation $$y'+xy=xy^2$$.

Solution

### 2419 solution video

video by MIP4U

$$\displaystyle{ xy'+y=\frac{1}{y^2} }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ xy'+y=\frac{1}{y^2} }$$.

Solution

### 2420 solution video

video by MIP4U

$$\displaystyle{ \frac{dy}{dx} + y = xy^4 }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ \frac{dy}{dx} + y = xy^4 }$$.

Solution

### 2421 solution video

$$\displaystyle{ \frac{dy}{dx} = \sqrt{y} - y; y(0) = 9 }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ \frac{dy}{dx} = \sqrt{y} - y; y(0) = 9 }$$.

Solution

### 2422 solution video

video by blackpenredpen

$$\displaystyle{ \frac{dy}{dx} - \frac{1}{x}y = xy^2 }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ \frac{dy}{dx} - \frac{1}{x}y = xy^2 }$$.

Solution

### 2423 solution video

$$\displaystyle{ y' + \frac{2}{x}y = -x^9y^5; y(1) = 1 }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ y' + \frac{2}{x}y = -x^9y^5; y(1) = 1 }$$.

Solution

### 2424 solution video

video by MIP4U

$$y' = -y(1 + xy^2)$$

Problem Statement

Solve the Bernoulli Equation $$y' = -y(1 + xy^2)$$.

Solution

### 2425 solution video

video by blackpenredpen

$$\displaystyle{ y' + \frac{4}{x} y = x^3y^2 }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ y' + \frac{4}{x} y = x^3y^2 }$$.

Solution

This problem was solved by three different people. We include all three videos here in case one helps you more than the other.

### 2426 solution video

video by MIP4U

$$y' + 2xy = xy^3$$

Problem Statement

Solve the Bernoulli Equation $$y' + 2xy = xy^3$$.

Solution

### 2427 solution video

$$\displaystyle{ y' + \frac{1}{t}y = -ty^3 }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ y' + \frac{1}{t}y = -ty^3 }$$.

Solution

### 2428 solution video

video by Engineer4Free

$$\displaystyle{ \frac{dy}{dt} = 2y + y^5; y(0) = 1 }$$

Problem Statement

Solve the Bernoulli Equation $$\displaystyle{ \frac{dy}{dt} = 2y + y^5; y(0) = 1 }$$.

Solution

video by MIP4U