Problem 3 - Calculus

Question

Separate, integrate, and solve equations \(1-8 .\) $$ d y / d x=x i y^{2}, \quad y_{0}=1 $$

Step-by-Step Solution

Answer

\(y = \frac{2}{2 - x^2}\).

Step 1: Separate Variables

The given differential equation is \( \frac{dy}{dx} = x \cdot y^2 \). To separate variables, we need to rearrange the equation so that all terms involving \(y\) are on one side and all terms involving \(x\) are on the other side. Divide both sides by \(y^2\) and multiply by \(dx\): \( \frac{1}{y^2} dy = x \, dx \).

Step 2: Integrate Both Sides

Integrate both sides of the equation. The left-hand side becomes: \( \int \frac{1}{y^2} \, dy = \int y^{-2} \, dy = -y^{-1} + C_1 \). The right-hand side becomes: \( \int x \, dx = \frac{x^2}{2} + C_2 \).

Step 3: Solve for the Constant

Combine the integration results and solve for the constant using the initial condition \( y_0 = 1 \). Substitute \( y = 1 \) and \( x = 0 \) into the integrated equation: \[ -y^{-1} = \frac{x^2}{2} + C \rightarrow -1 = 0 + C \rightarrow C = -1 \].

Step 4: Solve the Equation

Now substitute \(C = -1\) back into the integrated equation: \[ -y^{-1} = \frac{x^2}{2} - 1 \].Rearrange to find \(y\): \[ y^{-1} = 1 - \frac{x^2}{2} \] or equivalently, \[ y = \frac{1}{1 - \frac{x^2}{2}} = \frac{2}{2 - x^2} \].

Key Concepts

Variable Separation

Variable separation is a method used to solve differential equations by manipulating the equation into a form where all terms involving one variable are on one side, and all terms involving the other variable are on the opposite side. In the provided exercise, our goal is to isolate terms involving the variables so we can prepare the equation for integration.
To separate the variables in the differential equation \( \frac{dy}{dx} = x \cdot y^2 \), we aim to get all \(y\) terms on one side and all \(x\) terms on the other. We achieve this by dividing both sides by \(y^2\) and then multiplying through by \(dx\). This results in the equation:

• \( \frac{1}{y^2} \ dy = x \, dx \)

This form allows each side to be integrated separately, which is an essential step in solving differential equations using variable separation.

Integration

Integration is the process of finding the antiderivative, and it's crucial in solving differential equations once the variables have been separated.
After separating the equation as \( \frac{1}{y^2} \ dy = x \, dx \), we proceed to integrate both sides:

• The left side, \( \int \frac{1}{y^2} \, dy \), simplifies to \(-y^{-1} + C_1\).
• The right side, \( \int x \, dx \), simplifies to \( \frac{x^2}{2} + C_2 \).

After integration, we combine the results into a single equation representing the general solution of the differential equation. Integrating with respect to each variable separately is a standard practice that allows us to incorporate initial conditions later in the solving process.

Initial Conditions

Initial conditions are extra pieces of information provided in problems that allow us to find the specific solution rather than the general solution of a differential equation. These conditions help determine the constants of integration.
In this problem, the initial condition given is \( y_0 = 1 \). This means that when \( x = 0 \), \( y \) should also equal 1. By substituting these values into our equation from the integration step, we can solve for the constant \( C \):

• Substitute \( y = 1 \) and \( x = 0 \) into \(-y^{-1} = \frac{x^2}{2} + C \).
• We find \( -1 = 0 + C \), so \( C = -1 \).

Applying the initial conditions ensures that the solution satisfies not just the differential equation, but the specific context or scenario given by the problem.

Rearranging Equations

Rearranging equations is an important step in the process of solving differential equations. It involves simplifying or reshaping the equation to solve for the desired variable.
After integrating and applying the initial conditions in our problem, we arrive at the equation \(-y^{-1} = \frac{x^2}{2} - 1\). Our next goal is to solve for \(y\):

• Begin by removing the negative exponent, which gives us \(y^{-1} = \frac{1}{1 - \frac{x^2}{2}}\).
• Finally, take the reciprocal to express \(y\) as: \(y = \frac{2}{2 - x^2}\).

This rearranged form provides the explicit expression for \(y\) in terms of \(x\) and concludes the problem with the particular solution to the differential equation, tailored to the given initial condition.