91Ó°ÊÓ

An initial value problem involves a differential equation with an additional constraint called the initial condition. This typically specifies the value of the function, or its derivatives, at a certain point. Solving these problems means finding a function that satisfies both the differential equation and the initial condition.

In the initial value problem presented here, we have a differential equation \( (x^{2}+2 y^{2}) \frac{d x}{d y} = x y \) alongside the initial condition \( y(-1) = 1 \). The initial condition provides essential information to determine a specific solution from a family of possible solutions determined by the differential equation.

Understanding how to apply the initial condition after solving the differential equation is crucial. It will help you determine any constants that appear in the solution after integration.

Variable Separation is a method used to solve differential equations by separating the different variables on each side of the equation. This technique is very useful when dealing with equations that can be rearranged so that each variable and its differential can be placed as functions of each separate side.

To solve the differential equation \( (x^{2}+2 y^{2}) \frac{d x}{d y} = x y \) using variable separation:

• Rearrange the terms of the equation to separate the variables \(x\) and \(y\)
• Ensure that all terms involving \(x\) are on one side, and all terms involving \(y\) are on the other.

In our solution, we divided both sides by \(x\) and multiplied by \(dy\), resulting in \((x + \frac{2y^2}{x}) dx = y dy\). This separation allows us to integrate both sides separately to find the solution.

Once variables are separated, integration allows us to solve the equation. Each side of the equation becomes an integral that we have to solve.

For the differential equation \( (x + \frac{2y^2}{x}) dx = y dy \), we perform integration on both sides:

• The left-hand side simplifies to the integral \( \int (x + \frac{2y^2}{x}) \, dx \).
• The right-hand side becomes \( \int y \, dy \).

The integration on the left gives us \( \frac{x^2}{2} + 2y^2 \ln|x| \), and the right yields \( \frac{y^2}{2} \).

Always integrate with respect to the correct variable, taking care to include the constants of integration that arise, which will be reconciled using initial conditions in subsequent steps.

After integrating both sides of the differential equation, we introduce an integration constant, typically denoted as \(C\). However, this constant is not arbitrary.

We must use the initial condition given in the problem to find its value. For the problem at hand, we have \( y(-1) = 1 \). This means when \(x = -1\), \(y\) should equal 1. Substituting these values into the integrated equation \( \frac{x^2}{2} + 2y^2 \ln|x| = \frac{y^2}{2} + C \), will allow us to solve for \(C\).

• Input \(x = -1\) and \(y = 1\) into the equation.
• Solve to find \(C\).

Note that complications may arise if expressions such as \(\ln|-1|\) are encountered, which are undefined. This step might indicate a need to reassess earlier assumptions or consider the solution's domain.