Problem 33 - Differential Equations and Linear Algebra

Question

Find the general solution to the given differential equation and the maximum interval on which the solution is valid. $$y^{\prime}=\sin x$$.

Step-by-Step Solution

Answer

The general solution to the given differential equation \(y'(x) = \sin(x)\) is \(y(x) = -\cos(x) + C\), where C is the constant of integration. The maximum interval on which the solution is valid is \((-∞, ∞)\), as the function is defined for all real numbers and has no singularities, discontinuities, or other issues that would restrict its domain.

Step 1: Integrate the given differential equation

First, we need to integrate both sides of the given differential equation, \(y'(x) = \sin(x)\), with respect to x. \[y(x) = \int{y'(x) \, \mathrm{d}x} = \int{\sin(x) \, \mathrm{d}x}\]

Step 2: Find the antiderivative of \(\sin(x)\)

In order to evaluate the integral and find the general solution, we need to remember that the antiderivative of \(\sin(x)\) is \(-\cos(x)\). So the general solution will be: \[y(x) = -\cos(x) + C\] where C is the constant of integration.

Step 3: Determine the maximum interval on which the solution is valid

The general solution, \(y(x)=-\cos(x)+C\), is a function that is defined for all real numbers and it does not have any singularities, discontinuities, or other issues that would restrict its domain. The function is periodic with period \(2\pi\), so it repeats itself at regular intervals. Therefore, the maximum interval on which the general solution is valid is the entire real number line. The final answer is: General solution: \(y(x) = -\cos(x) + C\) Maximum interval of validity: \((-∞, ∞)\)

Key Concepts

General Solution

In the context of differential equations, the general solution encompasses all potential solutions of the equation. For the problem given, we start with the equation \(y'(x) = \sin(x)\). Integrating both sides, we look for a function \(y(x)\) whose derivative is \(\sin(x)\). This process yields a family of functions, identified by an arbitrary constant \(C\), which together form the general solution.
By integrating, we establish:

• \(y(x) = -\cos(x) + C\)

This form represents infinitely many solutions, each corresponding to a different value of \(C\), such as \(C = 0, 1, -1\), and so on.

Integration

Integration is a mathematical technique used to find an antiderivative. In the given problem, integrating the differential equation \(y'(x) = \sin(x)\) involves finding a function whose derivative is \(\sin(x)\). Integration effectively reverses differentiation.
To perform integration, we use the integral symbol \(\int\), similar to a summation of infinitely small quantities. In this exercise:

• The integration of \(\sin(x)\) with respect to \(x\) results in \(-\cos(x) + C\).
• Our task is to integrate each side of the differential equation to express \(y(x)\).

This process is crucial because it transforms the differential equation into a general solution, offering a complete family of solutions through the inclusion of the constant \(C\).

Antiderivative

The antiderivative of a function is a function whose derivative yields the original function. In simpler terms, it's the reverse of taking the derivative. Given the function \(\sin(x)\), we seek another function whose derivative is \(\sin(x)\).
For \(\sin(x)\), its antiderivative is \(-\cos(x)\), as the differentiation of \(-\cos(x)\) returns \(\sin(x)\). Thus,

• The antiderivative of \(\sin(x)\) is calculated as \(-\cos(x)\).
• It reflects the general solution by introducing the integration constant \(C\).

Knowing how to compute antiderivatives is a key skill for solving differential equations, allowing one to construct the general solution.

Interval of Validity

The interval of validity indicates where the solution to a differential equation is applicable. Evaluating the function \(y(x) = -\cos(x) + C\), we aim to determine the domain across the real number line where the solution holds without issues.
For this problem,

• \(-\cos(x)\) and hence \(y(x)\) is defined for all real numbers.
• The function is continuous and has no singularities or discontinuities. Thus, its domain encompasses all real numbers, \((-\infty, \infty)\).

This characteristic is essential for understanding the nature and applicability of solutions in real-world contexts, indicating that the solution remains valid without restrictions.