Question

Solving a First-Order Linear Differential Equation In Exercises \(5-14,\) solve the first-order linear differential equation. $$ y^{\prime}-3 x^{2} y=e^{x^{3}} $$

Short answer

The solution to the first-order linear differential equation is \(y = (x + c) * e^{x^3}\)

Step-by-step solution

Identify the Integrating Factor

In a first-order linear differential equation such as this, the integrating factor (IF) is \(e^{\int P(x)dx}\), where \(P(x)\) is the coefficient of the dependent variable in the differential equation. Here, \(P(x) = -3x^2\). So the IF is \(e^{\int -3x^2dx} = e^{-x^3}\).

Multiply Through By the Integrating Factor

With the IF identified, the next step is to multiply every term in the original differential equation by the IF. Therefore, multiplying each side of \(y' - 3x^2y = e^{x^3}\) by \(e^{-x^3}\), we have \(e^{-x^3}y' - 3x^2e^{-x^3}y = 1\).

Simplify and Express as a Derivative

We need to simplify the left hand side to be expressed as a derivative of a single function. The left hand side is a derivative \(d/dx[y * e^{-x^3}]\). Therefore, the derived equation is \(\frac{d}{dx}[y * e^{-x^3}]= 1\).

Integration

We integrate both sides of the equation \(\int d(y * e^{-x^3}) = \int dx\). This yields \(y * e^{-x^3} = x + c\), where c is the constant of integration.

Solve for y

Finally, we solve for y in the equation to find the solution to the differential equation. We therefore get \(y = (x + c) * e^{x^3}\).

Key concepts

Integrating Factor

In solving first-order linear differential equations, the Integrating Factor (IF) is a vital concept that can turn a difficult problem into a more manageable one. The IF is a function, often denoted by the Greek letter mu ((mu)), which, when multiplied by the original differential equation, simplifies it to a form that can be easily integrated. The general form for the IF in a first-order equation like \(y' + P(x)y = Q(x)\) is \(e^{\int P(x) dx}\).

For the problem \(y' - 3x^2y = e^{x^3}\), identifying the IF involves finding an antiderivative of the coefficient \(-3x^2\) of the dependent variable \(y\). This results in the IF \(e^{-x^3}\), which we multiply by the differential equation to simplify it. Essentially, it enables us to write the left side of the differential equation as the derivative of a product of functions, which is particularly convenient for taking the next steps towards finding the solution.

Differential Equation Solution

Solving a differential equation means finding a function or set of functions that satisfy the equation. The Differential Equation Solution refers to the expression or functions that represent the behavior of variables in relation to each other. In the case of a first-order linear differential equation, the solution will generally include a variable part and a constant part, represented by \(c\), known as the constant of integration.

In our example, after simplifying the equation and finding the integrating factor, the solution of the differential equation is expressed as \(y = (x + c) * e^{x^3}\), where \(x + c\) represents the antiderivative of the non-homogeneous part, with \(c\) accounting for all indefinite integrals' arbitrariness.

Constant of Integration

The Constant of Integration is an essential aspect of solving differential equations involving integration. This constant, typically denoted as \(c\), arises because the antiderivative of a function is not unique; rather, it represents a family of functions differing by a constant. When we integrate both sides of an equation, we introduce this constant to account for all possible antiderivatives of the original function.

For instance, when integrating the derivative to find the solution of our example equation, we get \(y * e^{-x^3} = x + c\). The \(c\) is critical because it allows for the representation of an infinite set of possible solutions, each corresponding to a different initial condition or different starting point on the graph of the solution.

Separation of Variables

Although the method of Separation of Variables is not directly applied in the solution of our example, it's an important technique worth understanding for other types of first-order differential equations. This method involves rearranging the differential equation to isolate different variables (often \(x\) and \(y\)) on opposite sides of the equation, allowing each side to be integrated separately.

In equations where separation of variables is applicable, you would typically divide both sides by the function of \(y\) and multiply by \(dx\) to get everything involving \(y\) on one side and everything involving \(x\) on the other. This results in an equation that can be integrated with respect to \(y\) on one side and \(x\) on the other, eventually leading to the solution of the differential equation. While our current example employs the integrating factor for a solution, understanding separation of variables can be useful for tackling a wide range of differential equations that you may encounter in your studies.