Question

Finding a General Solution In Exercises \(41-52,\) use integration to find a general solution of the differential equation. $$ \frac{d y}{d x}=x e^{x^{2}} $$

Short answer

The general solution to the differential equation \(\frac{dy}{dx} = xe^{x^2}\) is \(y = \frac{1}{2}e^{x^2} + C\).

Step-by-step solution

Identify the Integrating Function

First, identify the function that needs to be integrated. Here, the function is \(xe^{x^2}\). This is a simple power function multiplied by an exponential function.

Apply Integration

Since we're asked to solve \(\frac{dy}{dx} = xe^{x^2}\), to find y as a function of x, integrate both sides with respect to x. This means we have \(\int dy = \int xe^{x^2} dx\). The left hand side is trivially integrated to y. So, it is only necessary to solve the right hand integration.

Use Technique of Substitution

Using the technique of integration by substitution, let \(u=x^2\). Then, the derivative of \(u\) with respect to \(x\) is \(\frac{du}{dx} = 2x\). And \(\frac{du}{2} = x dx\). Change variables in the integral to get \( \int xe^{x^2} dx = \int \frac{e^u}{2} du\).

Evaluate the Integral

Solving the integral gives \( \frac{1}{2}\int e^u du\), which is simply \( \frac{1}{2}e^u + C\). Substitute back \(u = x^2\) to get \( \frac{1}{2}e^{x^2} + C\).

Final Solution

So the solution to the differential equation \(\frac{dy}{dx} = xe^{x^2}\) is \(y = \frac{1}{2}e^{x^2} + C\) where \(C\) represents constants of integration.

Key concepts

Integration

Integration is a fundamental concept in calculus that helps us find the antiderivative or the area under a curve. It is essentially the reverse of differentiation. When solving a differential equation, the goal is often to find a function whose derivative matches a given expression. This process involves integrating the expression.

• Consider the expression we need to integrate: \( \frac{dy}{dx} = xe^{x^2} \).
• This tells us to find the function \( y \) whose derivative is \( xe^{x^2} \).

In this exercise, the integration reveals the function \( y \) as our solution. Once integrated, we determine the integral of the right-hand side to find the general solution, which also includes a constant \( C \) representing unknown initial conditions.

Substitution Method

The substitution method is a powerful technique in integration that simplifies complex expressions. It works by changing variables to make the integral easier to solve. Here's how it's utilized in the exercise:

• You start by identifying parts of the integral that can transform into a simpler form.
• Here, we chose \( u = x^2 \) because it simplifies the exponential component \( e^{x^2} \).
• This substitution turns the integral \( \int xe^{x^2} dx \) into \( \int \frac{e^u}{2} du \).

After substitution, the integral becomes manageable, allowing for straightforward evaluation. Once integrated, the solution is converted back to the original variable to maintain consistency with the problem.

Exponential Functions

Exponential functions are mathematical functions of the form \( a^x \), where \( a \) is a constant. In the expression \( xe^{x^2} \), the exponential component is \( e^{x^2} \). These functions grow rapidly and are crucial in many real-world models, such as population growth and radioactive decay.

• The base of these exponential functions here is \( e \), an important constant approximately equal to 2.718.
• In calculus, the derivative and integral of exponential functions have predictable patterns, making them easier to work with than other non-linear functions.

Understanding the behavior of exponential functions helps recognize patterns in equations, aiding in solving differential equations like the one in this exercise.

Power Functions

Power functions are of the form \( x^n \), where \( n \) is a constant. In this exercise, the power function is embodying the \( x \) besides the exponential function. This component is necessary to the solution of the differential equation.

• These functions are characterized by straightforward differentiation and integration rules.
• Combining them with exponential terms can complicate the process, but these complications can be resolved using methods like substitution.

By recognizing power functions, you can predict how they behave when differentiated or integrated. This knowledge simplifies working through complex expressions and defining accurate solutions for differential equations.