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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.1.39
Chapter 9, Problem 9.1.39

33–42. Solving initial value problems Solve the following initial value problems.
y''(t) = teᵗ, y(0) = 0, y'(0) = 1

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Identify the given differential equation and initial conditions: \(y''(t) = t e^{t}\), with \(y(0) = 0\) and \(y'(0) = 1\).
Integrate the second derivative \(y''(t)\) once with respect to \(t\) to find the first derivative \(y'(t)\). This means computing \(y'(t) = \int t e^{t} \, dt + C_1\), where \(C_1\) is a constant of integration.
To integrate \(\int t e^{t} \, dt\), use integration by parts. Let \(u = t\) and \(dv = e^{t} dt\), then find \(du\) and \(v\), and apply the formula \(\int u \, dv = uv - \int v \, du\).
After finding \(y'(t)\), use the initial condition \(y'(0) = 1\) to solve for the constant \(C_1\).
Integrate \(y'(t)\) with respect to \(t\) to find \(y(t)\), adding another constant of integration \(C_2\). Then use the initial condition \(y(0) = 0\) to solve for \(C_2\).

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Key Concepts

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

Second-Order Differential Equations

A second-order differential equation involves the second derivative of a function. Solving such equations requires finding a function y(t) whose second derivative matches the given expression, often involving integration and applying initial conditions.
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Classifying Differential Equations

Initial Value Problems (IVPs)

An initial value problem specifies the values of a function and its derivatives at a particular point. These conditions allow us to determine the unique solution to a differential equation by solving for constants after integration.
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Initial Value Problems

Integration of Non-Homogeneous Terms

When the differential equation includes a non-homogeneous term like te^t, solving involves integrating this term twice. Techniques such as integration by parts are often used to handle products of functions like t and e^t.
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Divergence Test (nth Term Test)
Related Practice
Textbook Question

33–42. Solving initial value problems Solve the following initial value problems.

p'(x) = 2/(x² + x), p(1) = 0

Textbook Question

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.


Textbook Question

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y'(t) = eᵗʸ, y(0) = 1

Textbook Question

Explain why the graph of the solution to the initial value problem y'(t) = t²/(1 - t), y(-1) = ln 2 cannot cross the line t = 1.

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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.

(t² + 1)³yy'(t) = t(y² + 4)

Textbook Question

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁t⁵ + C₂t⁻⁴ - t³; t²u''(t) - 20u(t) = 14t³