Textbook Question
17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = y³sin t, y(0) = 1
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Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 9.1.30
21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
y'(t) = t lnt + 1
Verified step by step guidance1
Recognize that the given differential equation is a first-order ordinary differential equation of the form \(y'(t) = f(t)\), where \(f(t) = t \ln t + 1\).
To find the general solution, integrate both sides with respect to \(t\): \(y(t) = \int (t \ln t + 1) \, dt + C\), where \(C\) is an arbitrary constant.
Split the integral into two parts: \(\int t \ln t \, dt\) and \(\int 1 \, dt\).
Use integration by parts to solve \(\int t \ln t \, dt\). Let \(u = \ln t\) and \(dv = t \, dt\), then find \(du\) and \(v\) accordingly.
After computing the integrals, combine the results and add the constant \(C\) to write the general solution \(y(t)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ordinary Differential Equations (ODEs)
An ordinary differential equation relates a function and its derivatives with respect to one variable. Solving an ODE involves finding a function that satisfies the equation, often including arbitrary constants representing a family of solutions.
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Classifying Differential Equations
Integration of Functions
To solve first-order ODEs like y'(t) = f(t), we integrate the right-hand side with respect to t. Integration reverses differentiation and introduces an arbitrary constant, representing the general solution's family.
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Properties of Logarithmic Functions
Understanding the natural logarithm function ln(t) and its behavior is essential when integrating expressions like t ln(t). Techniques such as integration by parts are often used to integrate products involving logarithms.
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Related Practice
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) = C₁eᵗ + C₂teᵗ; u''(t) - 2u'(t) + u(t) = 0
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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.
y(t) = C₁ sin4t + C₂ cos4t; y''(t) + 16y(t) = 0
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Textbook Question
5–10. First-order linear equations Find the general solution of the following equations.
v'(y) − v/2 = 14
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Textbook Question
Orthogonal trajectories Use the method in Exercise 44 to find the orthogonal trajectories for the family of circles x² + y² = a²
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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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