Problem 12 - Calculus Single Variable

Question

Solve the given differential equation. $$ \frac{d y}{d x}+3 x y^{2}=0 $$

Step-by-Step Solution

Answer

The solution is \( y(x) = \frac{2}{3x^2 - 2C} \), with \( C \) as a constant.

Step 1: Identify the type of differential equation

The given differential equation is \( \frac{dy}{dx} + 3xy^2 = 0 \). It is a first-order, non-linear ordinary differential equation because of the \( y^2 \) term.

Step 2: Simplify the form

Rewrite the equation in the form \( \frac{dy}{dx} = -3xy^2 \). This isolates the derivative on one side.

Step 3: Separate variables

To separate variables, rewrite the equation as \( dy = -3x y^2 dx \).

Step 4: Perform separation of variables

We need to isolate \( y \) and \( x \) on opposite sides: move terms involving \( y \) to one side and \( x \) to the other side: \( \frac{1}{y^2} dy = -3x dx \).

Step 5: Integrate both sides

Integrate both sides separately: \( \int \frac{1}{y^2} dy = \int -3x dx \). The left integral becomes \( -\frac{1}{y} + C_1 \) and the right integral becomes \( -\frac{3}{2}x^2 + C_2 \).

Step 6: Combine the constants

Combine the integration constants: \( -\frac{1}{y} = -\frac{3}{2}x^2 + C \), where \( C = C_2 - C_1 \).

Step 7: Solve for \( y \)

To express \( y \): interchanging sides and solving for \( y \), we get \( y = \frac{1}{\frac{3}{2}x^2 - C} \). Therefore, \( y(x) = \frac{2}{3x^2 - 2C} \), where \( 2C \) is a new arbitrary constant.

Key Concepts

First-Order Differential Equation

A first-order differential equation is one of the simplest types of differential equations. It involves the derivative of the first degree (first derivative) of a function. For example, in the equation \( \frac{dy}{dx} + 3xy^2 = 0 \), the highest derivative present is \( \frac{dy}{dx} \). This qualifies it as a first-order equation.

• "First-order" simply means that the equation includes first derivatives and not higher derivatives like second or third.
• These equations often describe a wide range of simple systems in different fields such as physics, chemistry, and economics.

By understanding the order of a differential equation, you can predict the type of solutions you might encounter. This is the first step in identifying how to approach solving the equation.

Separation of Variables

Separation of variables is a technique used to solve some differential equations. It involves rearranging the equation so that all terms involving one variable appear on one side of the equation and all terms involving another variable appear on the opposite side.

• For the equation \( \frac{dy}{dx} = -3xy^2 \), separating variables involves rearranging it to \( \frac{1}{y^2} dy = -3x dx \).
• This allows us to integrate each side with respect to its respective variable.

This method is especially handy for first-order differential equations and is a straightforward way to make an equation easier to solve. Once variables are separated, the equation can be tackled using integration.

Non-Linear Differential Equation

A non-linear differential equation, like the one in this problem, involves terms where the dependent variable or its derivatives are raised to a power or multiplied together. In our equation, the term \( 3xy^2 \) makes it non-linear due to the \( y^2 \) factor.

• Non-linear equations are generally more complex than linear ones because of these power terms.
• They can result in a wider variety of behaviors and solutions which are not just combinations or multiples of known solutions.

Solving non-linear differential equations often requires specific techniques like separation of variables or numeric approximations because they don't always have straightforward solutions like linear ones.

Integration

Integration is the process of finding the integral of a function. When solving differential equations, once variables are separated, we integrate each side to find an antiderivative. In the example \( \int \frac{1}{y^2} dy = \int -3x dx \), we perform integration on both sides:

• On the left, the integral results in \( -\frac{1}{y} + C_1 \).
• On the right, the integral is \( -\frac{3}{2}x^2 + C_2 \).

Constants of integration \( C_1 \) and \( C_2 \) appear since the indefinite integrals don't specify the limits of integration.
Once integrated, these expressions are set equal and solved to express one variable in terms of another. This results in the solution to the original differential equation, allowing us to understand how \( y \) changes with respect to \( x \).