Textbook Question
Use Euler’s method with dx = 0.2 to estimate y(2) if y′ = y/x and y(1) = 2. What is the exact value of y(2)?
Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 9, Problem 9.2.32
Solve the Bernoulli equations in Exercises 29–32.
x²y' + 2xy = y³
Verified step by step guidance1
Rewrite the given differential equation \(x^{2}y' + 2xy = y^{3}\) in the standard Bernoulli form. First, isolate \(y'\) by dividing the entire equation by \(x^{2}\):
\[y' + \frac{2}{x}y = \frac{y^{3}}{x^{2}}\]
Identify the Bernoulli equation parameters: it has the form
\[y' + P(x)y = Q(x)y^{n}\]
where \(P(x) = \frac{2}{x}\), \(Q(x) = \frac{1}{x^{2}}\), and \(n = 3\).
Make the substitution \(v = y^{1-n} = y^{1-3} = y^{-2}\). Then compute \(v'\) in terms of \(y\) and \(y'\):
\[v = y^{-2} \implies v' = -2y^{-3}y'\]
Express \(y'\) from the substitution:
\[y' = -\frac{1}{2} y^{3} v'\]
Substitute \(y'\) and \(y^{3}\) back into the original equation rewritten in step 1 to get a linear differential equation in terms of \(v\).
After substitution, simplify the resulting equation to the form
\[v' + p(x)v = q(x)\]
where \(p(x)\) and \(q(x)\) are functions of \(x\). Then solve this linear first-order ODE using an integrating factor.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Bernoulli Differential Equation
A Bernoulli differential equation has the form y' + P(x)y = Q(x)y^n, where n is any real number other than 0 or 1. It is nonlinear due to the y^n term but can be transformed into a linear equation using an appropriate substitution.
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Classifying Differential Equations
Substitution Method for Bernoulli Equations
To solve a Bernoulli equation, use the substitution v = y^(1-n), which converts the nonlinear equation into a linear differential equation in terms of v. This allows the use of standard methods for linear equations to find the solution.
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07:33Euler's Method
Solving Linear First-Order Differential Equations
Once transformed, the equation becomes linear and can be solved using an integrating factor, μ(x) = exp(∫P(x) dx). Multiplying through by μ(x) simplifies the equation to an exact derivative, enabling integration and solution for v, and subsequently y.
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06:06Solving Separable Differential Equations
Related Practice
Textbook Question
Using Euler’s Method
In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.
y' = 2xy + 2y, y(0) = 3, dx = 0.2
Textbook Question
Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.
(x+1) dy/dx - 2 (x² + x)y = exp(x²) / (x+1), x > -1, y(0) = 5
Textbook Question
Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.
t dy/dt + 2y = t³, t > 0, y(2) = 1
Textbook Question
Show that the solution of the initial value problem
y' = x + y, y(x₀) = y₀
is
y = -1 -x + (1 + x₀ + y₀) exp(x-x₀).
Textbook Question
Use Euler’s method with dx = 0.5 to estimate y(5) if y′ = y²/√x and y(1) = −1. What is the exact value of y(5)?