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 9.4.42

Chapter 9, Problem 9.4.42

39–42. Special equations A special class of first-order linear equations have the form a(t)y'(t)+a'(t)y(t)=f(t), where a and f are given functions of t. Notice that the left side of this equation can be written as the derivative of a product, so the equation has the form
a(t)y'(t) + a'(t)y(t) = d/dt (a(t)y(t)) = f(t).
Therefore, the equation can be solved by integrating both sides with respect to t. Use this idea to solve the following initial value problems.

(t² + 1)y′(t) + 2ty = 3t², y(2) = 8

Verified step by step guidance

Recognize that the given differential equation is of the form \(a(t)y'(t) + a'(t)y(t) = f(t)\), where \(a(t) = t^2 + 1\) and \(f(t) = 3t^2\).

Rewrite the left side as the derivative of a product: \(\frac{d}{dt} \big(a(t) y(t)\big) = f(t)\), which means \(\frac{d}{dt} \big((t^2 + 1) y(t)\big) = 3t^2\).

Integrate both sides with respect to \(t\): \(\int \frac{d}{dt} \big((t^2 + 1) y(t)\big) dt = \int 3t^2 dt\).

After integration, express the solution as \((t^2 + 1) y(t) = \int 3t^2 dt + C\), where \(C\) is the constant of integration.

Use the initial condition \(y(2) = 8\) to substitute \(t=2\) and \(y=8\) into the equation to solve for \(C\), then write the explicit formula for \(y(t)\) by dividing both sides by \((t^2 + 1)\).

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Key Concepts

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

First-Order Linear Differential Equations

These are differential equations of the form y' + p(t)y = q(t), where p and q are functions of t. They can often be solved using integrating factors or by recognizing patterns that simplify the equation, such as rewriting the left side as a derivative of a product.

Product Rule and Recognizing Derivatives of Products

The product rule states that d/dt [a(t)y(t)] = a(t)y'(t) + a'(t)y(t). Identifying that the left side of the equation matches this derivative allows rewriting the differential equation in a simpler form, facilitating direct integration.

Solving Initial Value Problems by Integration

Once the equation is expressed as d/dt [a(t)y(t)] = f(t), integrating both sides with respect to t yields a(t)y(t) = ∫f(t) dt + C. Applying the initial condition y(t₀) = y₀ helps determine the constant of integration, providing a unique solution.

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Initial Value Problems

Related Practice

Textbook Question

25–28. Two steps of Euler’s method For the following initial value problems, compute the first two approximations u1 and u2 given by Euler’s method using the given time step.

y′(t) = 2−y, y(0) = 1; Δt = 0.1

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Textbook Question

23–26. Loan problems The following initial value problems model the payoff of a loan. In each case, solve the initial value problem, for t≥0 graph the solution, and determine the first month in which the loan balance is zero.

B′(t) = 0.004B − 800, B(0) = 40,000

Textbook Question

Does the function y(t) = 2t satisfy the differential equation y'''(t) + y'(t) = 2?

Textbook Question

11–16. Initial value problems Solve the following initial value problems.

y'(x) = −y + 2, y(0) = −2

Textbook Question

45–48. General first-order linear equations Consider the general first-order linear equation y'(t)+a(t)y(t)=f(t). This equation can be solved, in principle, by defining the integrating factor p(t)=exp(∫a(t)dt). Here is how the integrating factor works. Multiply both sides of the equation by p (which is always positive) and show that the left side becomes an exact derivative. Therefore, the equation becomes

p(t)(y′(t) + a(t)y(t)) = d/dt(p(t)y(t)) = p(t)f(t).

Now integrate both sides of the equation with respect to t to obtain the solution. Use this method to solve the following initial value problems. Begin by computing the required integrating factor.

y′(t) + (2t)/(t² + 1)y(t) = 1 + 3t², y(1) = 4

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Textbook Question

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.

u'(x) = e²ˣ⁻ᵘ