Question

Find the general solution to each differential equation. $$\left(1+x^{3}\right) d y=\left(1-3 x^{2} y\right) d x$$

Short answer

The general solution involves integrating each side and then solving the resulting integrals to find a relationship between x and y that includes an arbitrary constant.

Step-by-step solution

Express the equation in differential form

Write the given differential equation expressing differentials of y and x on each side: \( (1+x^3)dy = (1-3x^2y)dx \).

Separation of Variables

Rearrange the terms to separate the variables x and y to the two sides of the equation: \( \frac{1}{1-3x^2y} dy = \frac{1}{1+x^3} dx \).

Integrate both sides

Integrate both sides of the equation to find the general solution. The integration will involve the antiderivatives of the respective sides. \( \int \frac{1}{1-3x^2y} dy = \int \frac{1}{1+x^3} dx \).

Solve the integrals

Compute the indefinite integrals on both sides. The integrals may not necessarily result in elementary functions, and thus, the explicit general solution can be expressed in terms of an integral if necessary.

Write the general solution

Express the general solution of the differential equation, incorporating the constants of integration from both sides.

Key concepts

General Solution of Differential Equations

Understanding the general solution of differential equations is a foundational skill in mathematics, particularly in fields such as physics, engineering, and economics. A differential equation is a mathematical equation that relates some function with its derivatives.

The term 'general solution' refers to a solution that includes all possible solutions to a differential equation. It usually contains arbitrary constants; for a first-order differential equation, there will be one constant. In the case of a higher-order differential equation, there could be more. For instance, a second-order differential equation will typically have two constants. These constants represent the 'initial conditions' that can be applied to find a specific (or particular) solution.

When you encounter a differential equation, like \( (1+x^3)dy = (1-3x^2y)dx \), the goal is to find a general solution that represents every possible antiderivative. The process often involves integration, as you can see in the step-by-step solution provided, where the final answer is an implicit function of both variables along with an integration constant.

Separation of Variables

Separation of variables is a technique used to solve a particular type of differential equation. It involves rearranging the equation to isolate each variable with its corresponding differential on either side of the equation. This technique works for equations where the variables can be separated in such a manner.

The method is based on the premise that if two functions are multiplied together to give a constant, then each function must separately be a constant. Thus, if we can write \( f(x)g(y) = C \) where C is a constant, then \( f(x) = C_1 \) and \( g(y) = C_2 \) for constants \( C_1 \) and \( C_2 \) which could be equal or multiples of each other.

In our example, the differential equation \( (1+x^3)dy = (1-3x^2y)dx \) is manipulated to isolate terms involving y on one side and x on the other before integrating, as shown: \( \frac{1}{1-3x^2y} dy = \frac{1}{1+x^3} dx \). Essentially, we're preparing the equation for the integration process to find the general solution.

Integration of Functions

Integration is a fundamental concept in calculus that is used to find areas, volumes, central points, and many useful things, but it is also crucial when dealing with differential equations.

Integration allows us to find the antiderivatives of functions. In the context of a differential equation, once variables are separated, we proceed to integrate both sides to find a solution. Integrals can be definite (with limits) or indefinite (without limits). In solving differential equations, we most commonly deal with indefinite integrals, which introduce a constant of integration due to the infinite number of antiderivatives for any given function.

After separating the variables in our example, integration is applied: \( \int \frac{1}{1-3x^2y} dy = \int \frac{1}{1+x^3} dx \). The ability to integrate these expressions, whether they result in elementary functions or not, is an important part of determining the general solution. The solution is then expressed along with the constants of integration which capture all of the possible antiderivatives corresponding to the given differential equation.