Problem 47
Question
In Problems 47-58, find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{2}{x}-x, x>0 $$
Step-by-Step Solution
Verified Answer
The general solution is \( y = 2\ln|x| - \frac{x^2}{2} + C \), where \( C \) is an arbitrary constant.
1Step 1: Identify the Type of Differential Equation
The given differential equation is \( \frac{d y}{d x} = \frac{2}{x} - x \). This is a first-order differential equation, and it can be categorized as a separable differential equation since the right-hand side can be expressed as a function of \( x \) alone.
2Step 2: Integrate to Find the General Solution
To find the general solution, we integrate both sides with respect to \( x \). Separately integrate the terms \( \frac{2}{x} \) and \( -x \). The integral of \( \frac{2}{x} \) is \( 2\ln|x| \) and the integral of \( -x \) is \( -\frac{x^2}{2} \). Thus, the general solution becomes \( y = 2\ln|x| - \frac{x^2}{2} + C \), where \( C \) is the constant of integration.
Key Concepts
First-order Differential EquationsSeparable Differential EquationsIntegrationGeneral Solution
First-order Differential Equations
A first-order differential equation involves derivatives of the first degree. These equations typically have the form \( \frac{dy}{dx} = f(x, y) \), where \( f(x, y) \) can be any function of \( x \) and \( y \). The primary goal when dealing with these equations is to find a function \( y(x) \) such that the equation holds true for all \( x \).
For example, given the differential equation \( \frac{dy}{dx} = \frac{2}{x} - x \), the equation is first-order because it involves the first derivative of \( y \) with respect to \( x \). In practice, solving first-order differential equations often requires simplifying the equation or rewriting it in a more workable form, such as turning it into a separable differential equation.
For example, given the differential equation \( \frac{dy}{dx} = \frac{2}{x} - x \), the equation is first-order because it involves the first derivative of \( y \) with respect to \( x \). In practice, solving first-order differential equations often requires simplifying the equation or rewriting it in a more workable form, such as turning it into a separable differential equation.
Separable Differential Equations
Separable differential equations are a specific type of first-order differential equation where the equation can be broken down into two parts: one exclusively involving \( y \) and the other involving \( x \). This ability to separate the variables is what makes these equations easier to solve.
To solve the differential equation \( \frac{dy}{dx} = \frac{2}{x} - x \), we see that the right-hand side is already expressed as a function of \( x \) alone, which simplifies the separation process.
To solve the differential equation \( \frac{dy}{dx} = \frac{2}{x} - x \), we see that the right-hand side is already expressed as a function of \( x \) alone, which simplifies the separation process.
- Identify parts: In this case, \( \frac{2}{x} \) and \( -x \) are functions of \( x \).
- Integrate both sides: This allows us to find \( y \) in terms of \( x \).
Integration
Integration is the mathematical process used to find the original function from its derivative. In the context of solving differential equations, integration is used to obtain the solution by reversing differentiation.
For the equation \( \frac{d y}{d x} = \frac{2}{x} - x \), we look to integrate both terms on the right side:
For the equation \( \frac{d y}{d x} = \frac{2}{x} - x \), we look to integrate both terms on the right side:
- Integrating \( \frac{2}{x} \) gives us \( 2\ln|x| \), because the integral of \( \frac{1}{x} \) is \( \ln|x| \).
- Integrating \( -x \) results in \( -\frac{x^2}{2} \), using the power rule for integration which states \( \int x^n dx = \frac{x^{n+1}}{n+1} \).
General Solution
The general solution of a differential equation represents an entire family of functions, capturing all possible solutions based on the equation given.
After integrating, we found that the solution for the differential equation \( \frac{dy}{dx} = \frac{2}{x} - x \) is \( y = 2\ln|x| - \frac{x^2}{2} + C \).
After integrating, we found that the solution for the differential equation \( \frac{dy}{dx} = \frac{2}{x} - x \) is \( y = 2\ln|x| - \frac{x^2}{2} + C \).
- \( C \) represents the constant of integration, accounting for the family of solutions.
- This solution is general because without initial conditions or specifics, that constant \( C \) can take any real value.
Other exercises in this chapter
Problem 46
In Problems \(41-46\), assume that \(a\) is a positive constant. Find the general antiderivative of the given function. $$ f(x)=\frac{e^{-a x}+e^{a x}}{2 a} $$
View solution Problem 46
Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Show that if \(f(a)
View solution Problem 47
Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). Assume further that \(f(a)=f(b)=0\) but \(f\) is not constant on \([a, b] .\) Ex
View solution Problem 48
Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty}\left(1+\frac{3}{x^{2}}\right)^{x} $$
View solution