Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.3.53b

Chapter 9, Problem 9.3.53b

Blowup in finite time Consider the initial value problem y'(t) = yⁿ + 1, y(0) = y₀, where n is a positive integer.
b. Solve the initial value problem with n = 2 and y₀ = 1/√2.

Verified step by step guidance

Start with the given initial value problem: \(y'(t) = y^2 + 1\) with initial condition \(y(0) = \frac{1}{\sqrt{2}}\).

Rewrite the differential equation in separable form: \(\frac{dy}{dt} = y^2 + 1\) implies \(\frac{dy}{y^2 + 1} = dt\).

Integrate both sides: integrate \(\int \frac{dy}{y^2 + 1}\) on the left and \(\int dt\) on the right. Recall that \(\int \frac{dy}{y^2 + 1} = \arctan(y) + C\).

After integration, write the implicit solution: \(\arctan(y) = t + C\). Use the initial condition \(y(0) = \frac{1}{\sqrt{2}}\) to solve for the constant \(C\) by substituting \(t=0\) and \(y=\frac{1}{\sqrt{2}}\).

Finally, solve for \(y\) explicitly by taking the tangent of both sides: \(y = \tan(t + C)\), which gives the solution to the initial value problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Separable Differential Equations

A separable differential equation can be written as dy/dt = g(y)h(t), allowing variables y and t to be separated on opposite sides of the equation. This enables integration with respect to each variable independently, which is essential for solving the given initial value problem.

Initial Value Problem (IVP)

An initial value problem specifies a differential equation along with a condition y(t₀) = y₀. This condition allows determination of the particular solution from the family of general solutions by solving for the integration constant.

Finite Time Blowup

Finite time blowup occurs when the solution to a differential equation becomes unbounded in a finite time interval. For nonlinear equations like y' = y² + 1, solutions can grow rapidly and approach infinity at a finite time, which is important to analyze after solving the IVP.

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b. Verify that M(0) = M₀

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. Euler’s method is used to compute exact values of the solution of an initial value problem.

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. The solution of a stirred tank initial value problem always approaches a constant as t→∞

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38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

b. Sketch the direction field, for t≥0.

y′(t) = 2y + 4

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17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

b. In what regions are solutions increasing? Decreasing?

y'(t) = y(y+3)(4-y)

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Textbook Question

b. Sketch the direction field, for t≥0.

y′(t) = 6 - 2y

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