Question

Find the general solution to the differential equation.\(y^{\prime}=2 x y^{2}\)

Short answer

The general solution is \( y = \frac{1}{-x^2 - C} \).

Step-by-step solution

Separate Variables

The given differential equation is \( y^{\prime} = 2xy^2 \). Start by rewriting the differential equation by separating the variables. This involves arranging all the \( y \) terms on one side and \( x \) terms on the other side:\[ \frac{dy}{y^2} = 2x \, dx \]

Integrate Both Sides

Integrate both sides of the equation from Step 1. For the left side, the integral of \( \frac{dy}{y^2} \) is \( -\frac{1}{y} \). For the right side, the integral of \( 2x \, dx \) is \( x^2 \):\[ \int \frac{dy}{y^2} = \int 2x \, dx \]This results in:\[ -\frac{1}{y} = x^2 + C \]

Solve for y

From the result of the integration, solve for \( y \) in terms of \( x \) and the constant \( C \). First, multiply both sides by \(-1\):\[ \frac{1}{y} = -x^2 - C \]Then, take the reciprocal to solve for \( y \):\[ y = \frac{1}{-x^2 - C} \]

Write the General Solution

The general solution of the differential equation is the expression obtained after solving for \( y \). Therefore, the general solution is:\[ y = \frac{1}{-x^2 - C} \] where \( C \) is an arbitrary constant.

Key concepts

Separation of Variables

One efficient method to solve differential equations is the separation of variables technique. This approach simplifies an equation by separating it into two distinct parts: one that only contains the variable \( y \) and its derivatives, and another that involves the variable \( x \).
For instance, let's consider the equation \( y^{\prime} = 2xy^2 \). The goal is to rearrange this equation so that terms with \( y \) are on one side and those with \( x \) are on the other.
This results in the separated form \( \frac{dy}{y^2} = 2x \cdot dx \). Here, we can clearly see that the left side will only deal with \( y \) and the right with \( x \).

• By separating the variables, we prepare the equation for integration.
• This technique simplifies the integration process, allowing for each side to be integrated individually.

General Solution

The general solution of a differential equation is the solution that includes arbitrary constants—denoted often as \( C \). These constants utilize initial conditions or boundary values to provide specific solutions to particular scenarios.
Consider the separated equation after integration: \( -\frac{1}{y} = x^2 + C \). This equation is the basis from which we determine the general solution.

• The presence of \( C \) indicates that there is an entire family of solutions.
• Each value of \( C \) corresponds to a specific solution depending on the initial conditions provided.

Solving for \( y \) gives the general solution: \( y = \frac{1}{-x^2 - C} \). This expression will satisfy the original differential equation for any given \( C \).
Thus, the general solution is a crucial part of understanding the behavior of the system described by the differential equation.

Integration

Integration is a fundamental concept used to solve differential equations, especially when dealing with separated variables. Integration involves finding the antiderivative of a function, which, in this context, helps us solve the differential equation.
Applying integration to the equation \( \frac{dy}{y^2} = 2x \cdot dx \) involves integrating each side individually.

Left Side Integration

The left side, \( \int \frac{dy}{y^2} \), requires finding the antiderivative of \( \frac{1}{y^2} \), which results in \( -\frac{1}{y} \).

Right Side Integration

The right side, \( \int 2x \cdot dx \), involves a simple power rule integration, giving us \( x^2 \).

• Each side must be treated separately according to its respective variables.
• Integration adds an arbitrary constant \( C \) to the results since integration is an inverse process of differentiation.

Therefore, after integrating, we obtain \( -\frac{1}{y} = x^2 + C \), which is a crucial step toward finding the general solution. Integration is not just a mathematical process but also a step toward understanding the relationship between variables and their changes over time.