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Magazine Chapter 1: Problem 16
\(d y / d x=x e^{-x^{2}}\)
Short Answer
Expert verified
The solution is \( y = -\frac{1}{2} e^{-x^2} + C \).
Step by step solution
01
- Identify the integral
To solve the differential equation \( \frac{dy}{dx} = xe^{-x^2} \), recognize that it requires integration to find the general solution.
02
- Set up the integral
Rewrite the equation in integral form: \[ y = \int xe^{-x^2} dx \]
03
- Use substitution method
To integrate \( xe^{-x^2} \), use the substitution \( u = -x^2 \), hence \( du = -2x dx \). This implies \( -\frac{1}{2} du = x dx \).
04
- Substitute into the integral
Replace \( x dx \) in the integral with \( -\frac{1}{2} du \), thus transforming the integral: \[ y = \int e^u \left(-\frac{1}{2} du\right) = -\frac{1}{2} \int e^u du \]
05
- Integrate
Integrate \( -\frac{1}{2} \int e^u du \) to get \[ y = -\frac{1}{2} e^u + C \]
06
- Substitute back the value of u
Replace \( u \) back with \( -x^2 \) to get the solution: \[ y = -\frac{1}{2} e^{-x^2} + C \]
07
- State the general solution
The general solution to the differential equation \( \frac{dy}{dx} = xe^{-x^2} \) is: \[ y = -\frac{1}{2} e^{-x^2} + C \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
integration
Integration is a key concept in solving differential equations. It allows us to find a function given its derivative. In this particular problem, the differential equation is \( \frac{dy}{dx} = xe^{-x^2} \). We start by rearranging this equation into the integral form: \( y = \int xe^{-x^2} dx \). Integration helps us reverse the process of differentiation, giving us the antiderivative which is crucial for finding our general solution. Always remember, if you're given a derivative and need to find the original function, think about integration.
substitution method
The substitution method simplifies complex integrals by changing variables. Here, we use the substitution \( u = -x^2 \), making the differentiation part manageable. This changes our expression from \( \int xe^{-x^2} dx \) to \( \int e^u \left(-\frac{1}{2} du\right) \). By substituting, we reduce the problem to a more straightforward integration. Remember the following steps for substituting:
- Choose a substitution that simplifies the integral
- Find the differential of your substitution (\( du, dx \) etc.)
- Replace all variables and differentials in the integral
general solution
The general solution incorporates all possible solutions to a differential equation. After integrating and obtaining \( y = -\frac{1}{2} e^u + C \), we replace \( u \) back with \( -x^2 \), resulting in \( y = -\frac{1}{2} e^{-x^2} + C \).
The constant \( C \) represents an arbitrary constant included because the antiderivative of a function is not unique; it can shift up or down by any constant value since the derivative of a constant is zero. Hence, the general solution \( y = -\frac{1}{2} e^{-x^2} + C \) captures all possible solutions to the differential equation. Every differential equation of the first order will have a general solution containing a constant like this.
The constant \( C \) represents an arbitrary constant included because the antiderivative of a function is not unique; it can shift up or down by any constant value since the derivative of a constant is zero. Hence, the general solution \( y = -\frac{1}{2} e^{-x^2} + C \) captures all possible solutions to the differential equation. Every differential equation of the first order will have a general solution containing a constant like this.
differential equation
A differential equation involves an unknown function and its derivatives. For this exercise, the given differential equation is \( \frac{dy}{dx} = xe^{-x^2} \).
Solving it involves several steps:
Differential equations describe various physical and real-world phenomena like motion, growth, and decay. Understanding how to solve them opens up the study of many dynamic systems and mathematical modeling of real-life situations.
Solving it involves several steps:
- Identifying the type of differential equation
- Manipulating and simplifying it
- Using integration and sometimes special techniques like substitution
Differential equations describe various physical and real-world phenomena like motion, growth, and decay. Understanding how to solve them opens up the study of many dynamic systems and mathematical modeling of real-life situations.
