Problem 6 - Advanced Engineering Mathematics

Question

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}+2 x y^{2}=0 $$

Step-by-Step Solution

Answer

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

Step 1: Rearrange Equation

Start by rearranging the differential equation \( \frac{d y}{d x} + 2xy^2 = 0 \) to isolate terms involving \( y \) and \( x \). We can rewrite this equation as \( \frac{d y}{d x} = -2xy^2 \).

Step 2: Separate Variables

Separate the variables by expressing all terms involving \( y \) on one side and all terms involving \( x \) on the other. This gives us \( \frac{1}{y^2} \, dy = -2x \, dx \).

Step 3: Integrate Both Sides

Integrate both sides of the equation. The integral of \( \frac{1}{y^2} \, dy \) is \( -\frac{1}{y} \), and the integral of \( -2x \, dx \) is \( -x^2 \). Therefore, the equation becomes \( -\frac{1}{y} = -x^2 + C \), where \( C \) is the constant of integration.

Step 4: Solve for y

Solve the implicit equation for \( y \). First, multiply through by -1 to get \( \frac{1}{y} = x^2 - C \). Taking the reciprocal gives us \( y = \frac{1}{x^2 - C} \).

Key Concepts

Separation of Variables

Separation of variables is a fundamental technique used to solve differential equations. When we encounter a differential equation like \( \frac{d y}{d x} + 2xy^2 = 0 \), our first goal is to rearrange the equation in a way that all terms involving \( y \) wind up on one side, while all terms related to \( x \) are isolated on the other. This rearrangement helps in addressing complex equations in a simpler way. Applying separation of variables to our equation, we first rearrange it so that the derivative \( \frac{d y}{d x} \) is isolated:

• Subtract \( 2xy^2 \) from both sides, resulting in \( \frac{d y}{d x} = -2xy^2 \).

The next step is taking terms associated with \( y \) (in this case, \( y^2 \)) to one side of the equation, and terms involving \( x \) to the other. We achieve this by dividing through by \( y^2 \) and multiplying by \( dx \), which gives us:

• \( \frac{1}{y^2} \ dy = -2x \ dx \)

This equation is now separated, meaning each variable stands clearly on its own, setting the stage for the integration process.

Integrating

Once we have our variables separated, the next logical step is to integrate each side. Integration is the process of finding the antiderivative, which will help us find a general solution for the original differential equation.In our separated equation \( \frac{1}{y^2} \ dy = -2x \ dx \), we proceed as follows:

• Integrate the left side with respect to \( y \): \( \int \frac{1}{y^2} \, dy \). This integral results in \( -\frac{1}{y} \).
• Integrate the right side with respect to \( x \): \( \int -2x \, dx \). This gives us \( -x^2 + C \), where \( C \) is the arbitrary constant of integration.

Integrating is crucial because it helps convert differential equations, which describe rates of change, into algebraic equations, which we can solve more directly. It’s the bridge from dynamic descriptions of phenomena to more static expressions.

Solving Differential Equations

After integrating both sides, we often find ourselves with an implicit equation that includes the constant of integration, \( C \). At this point, our task is to express \( y \) explicitly in terms of \( x \) or vice versa, if possible.In the integrated form we obtained, \( -\frac{1}{y} = -x^2 + C \), we can solve for \( y \) by first multiplying the entire equation by \(-1\):

• \( \frac{1}{y} = x^2 - C \)

Then, taking the reciprocal on both sides will allow us to isolate \( y \):

• From \( \frac{1}{y} = x^2 - C \), we derive \( y = \frac{1}{x^2 - C} \)

This gives us the explicit solution for \( y \) in terms of \( x \), completing the process of solving the differential equation. Differential equation solutions often include a constant \( C \) because there are many functions that can satisfy a given differential equation due to the integral process.