Question

Find the general solution to the differential equations. $$ y^{\prime}=x^{2}+3 e^{x}-2 x $$

Short answer

The general solution is \( y = \frac{x^3}{3} + 3e^x - x^2 + C \).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is \( y' = x^2 + 3e^x - 2x \). This is a first-order differential equation and can be solved by integrating both sides.

Integrate to Find the General Solution

To find the general solution, integrate both sides with respect to \( x \):\[ y = \int (x^2 + 3e^x - 2x) \, dx. \]

Integrate Each Term Separately

Break down the integration into three terms:1. \( \int x^2 \, dx = \frac{x^3}{3} + C_1 \)2. \( \int 3e^x \, dx = 3e^x + C_2 \)3. \( \int -2x \, dx = -x^2 + C_3 \)Where \( C_1, C_2, \) and \( C_3 \) are constants of integration.

Combine the Integrals

Combine the integrated terms to express the general solution:\[ y = \frac{x^3}{3} + 3e^x - x^2 + C, \]where \( C = C_1 + C_2 + C_3 \) represents an arbitrary constant of integration.

Key concepts

First-order Differential Equations

First-order differential equations are a fundamental type of differential equations where the highest derivative involved is the first derivative. In the context of the example given, the differential equation is presented as:

• \( y' = x^2 + 3e^x - 2x \)

This equation relates the rate of change of a function \( y \) with respect to its independent variable \( x \). Here, \( y' \) denotes the first derivative of \( y \) concerning \( x \). These types of equations often describe various natural phenomena, including population growth and radioactive decay, among other processes.
To solve such an equation, we need to integrate to find the original function \( y \). First-order differential equations can vary greatly in complexity, but this particular equation is linear in form, which allows integration as a viable method for finding solutions.

Integration

Integration is a fundamental mathematical process used to find a function from its derivative. In solving differential equations like the one at hand, integrating helps us find the general solution. Given our equation:

• \( y' = x^2 + 3e^x - 2x \)

We integrate each term separately to determine the antiderivative or the original function \( y \). Breaking down the integration process, we have:

• The integral of \( x^2 \) yields \( \frac{x^3}{3} \).
• The integral of \( 3e^x \) results in \( 3e^x \).
• The integral of \( -2x \) gives \( -x^2 \).

Each of these integrals involves adding a constant of integration, which we'll discuss in detail later. Integration allows us to piece together these individual results to articulate the general solution for the original differential equation.

General Solution

The term "general solution" refers to the set of all possible solutions to a differential equation. For first-order differential equations, the general solution takes into account an arbitrary constant resulting from the integration process. In our given example, integrating both sides of the equation provides:\[ y = \frac{x^3}{3} + 3e^x - x^2 + C \]Here, \( C \) represents an arbitrary constant called the constant of integration. This solution expresses \( y \) as a function of \( x \) and the constant \( C \). It illustrates how the solutions can shift vertically depending on the initial conditions or additional information provided.
The general solution is vital as it encompasses the entire family of solutions for the equation, providing us with flexibility to address specific scenarios and constraints that can come up in practical applications.

Constants of Integration

In the process of integrating, a constant of integration arises because differentiating a constant results in zero, and thus its presence is not seen in the original derivative. In the integration of our differential equation's individual terms, you'll encounter constants like \( C_1 \), \( C_2 \), and \( C_3 \).When these are combined, as shown in the step-by-step solution, they form a single constant \( C \) in the general solution:

• \( y = \frac{x^3}{3} + 3e^x - x^2 + C \)

This single constant \( C = C_1 + C_2 + C_3 \) indicates that, although the specific values of these constants might differ, together they define the space of potential solutions. The constant of integration is important for the feasibility and adaptability of solutions, making it possible to accommodate real-world constraints or known initial conditions, allowing the general solution to flex and fit particular scenarios.