Find the "general solution" (that is, a solution containing...

Find the "general solution" (that is, a solution containing an arbitrary constant) of each of the following differential equations, by separation of variables. Then find a particular solution of each equation satisfying the given boundary conditions. $\left(1+y^{2}\right) d x+x y d y=0$ $y=0$ when $x=5$

Find the "general solution" (that is, a solution containing an arbitrary constant) of each of the following differential equations, by separation of variables. Then find a particular solution of each equation satisfying the given boundary conditions.
$\left(1+y^{2}\right) d x+x y d y=0$ $y=0$ when $x=5$

Key Concepts

Separation of Variables

This concept involves rewriting a differential equation in such a way that all terms involving one variable (and its differential) appear on one side, while all terms involving the other variable appear on the opposite side. In doing so, the equation becomes integrable, allowing each side to be integrated with respect to its own variable, leading to the general solution.

Integration Techniques

In solving separated equations, integration techniques such as substitution are often employed to simplify the process. This involves identifying an appropriate substitution to handle non-trivial integrals, thereby transforming the integrals into a more manageable form and facilitating the solution of the differential equation.

General Solution and Arbitrary Constants

The general solution of a differential equation is obtained after integrating and contains one or more arbitrary constants. These constants represent the family of all possible solutions to the differential equation and are a direct result of the integration process.

Initial Conditions and Particular Solutions

Initial conditions, sometimes referred to as boundary conditions, are specific values provided for the variables that allow one to determine the arbitrary constants in the general solution. Once these constants are identified, the general solution becomes a particular solution that satisfies the given initial or boundary conditions.

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