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Chapter 10: Problem 2

Solve \(y^{\prime}=1 /\left(1+t^{2}\right)\).

Short Answer

Expert verified
The solution is \( y = \arctan(t) + C \).

Step by step solution

01

Identify the Type of Differential Equation

The given differential equation is \( y' = \frac{1}{1 + t^2} \). This is a first-order, separable differential equation that can be solved by direct integration of the right-hand side.
02

Integrate Both Sides

Integrate the right-hand side with respect to \( t \):\[y = \int \frac{1}{1 + t^2} dt\]This integral is a standard form and equals \( \arctan(t) + C \), where \( C \) is the constant of integration.
03

Write the General Solution

The solution to the differential equation is:\[y = \arctan(t) + C\]where \( C \) is an arbitrary constant, representing the family of solutions.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

First-Order Differential Equations
First-Order Differential Equations are a foundational concept in calculus and differential equations. They describe how a quantity changes with respect to another variable. In mathematical terms, they involve a derivative, such as \( y' \), which represents the rate of change of \( y \) in relation to the variable \( t \).

In the context of separable differential equations:
  • The equation can often be rewritten in a form where the two variables can be separated onto different sides.
  • This separation allows each side to be integrated individually, leading to a general solution.
  • Separable equations are typically written in the form \( \frac{dy}{dt} = g(t) \cdot h(y) \), where the functions of \( t \) and the functions of \( y \) can be dealt with separately.
To solve first-order differential equations, grasping these fundamentals can significantly simplify the process.
Integration
Integration is a central technique in calculus. It is the process of finding the integral, which can be thought of as the reverse operation of differentiation.

When dealing with the exercise provided, integration involves the following:
  • Recognizing the integral form to match standard integrals for easier computation.
  • The integral of \( \frac{1}{1 + t^2} \) involves a well-known result, making it simpler to evaluate.
The process of integration gives rise to a constant of integration, often denoted as \( C \). This constant represents a family of possible solutions and arises because the derivative of a constant is zero, meaning that different constants added to the function yield the same derivative.
Arctangent Function
The Arctangent Function, denoted as \( \arctan(t) \), is the inverse of the tangent function. It plays a significant role in integration, especially in problems involving quadratic expressions in the denominator.

Important aspects of the arctangent function include:
  • Its derivative, \( \frac{d}{dt}[\arctan(t)] = \frac{1}{1 + t^2} \), is crucial, as seen in the given differential equation.
  • Understanding this derivative greatly aids in identifying the integral as the arctangent function.
  • It is defined for all real numbers and has a range between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
These points make the arctangent a common result in differential equations where expressions like \( 1 + t^2 \) appear in integration problems.

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Most popular questions from this chapter

Solve the initial value problem. $$y^{\prime}+y \cos \left(e^{t}\right)=0, y(0)=0$$

In the following exercises, compute the Euler approximations for the initial value problem for \(0 \leq t \leq 1\) and \(\Delta t=0.2 .\) If you have access to Sage, generate the slope field first and attempt to sketch the solution curve. Then use Sage to compute better approximations with smaller values of \(\Delta t\). $$y^{\prime}=t \ln y, y(0)=2$$

Find the solution for the mass-spring equation \(y^{\prime \prime}+4 y^{\prime}+29 y=689 \cos (2 t)\).

Find the general solution to the differential equation using variation of parameters. $$y^{\prime \prime}+y=\tan x$$

Find the general solution to the differential equation. $$y^{\prime \prime}+9 y=3 \sin (3 t)$$
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