Solve the given differential equations. √(x^2+y^2) d x-2 y d y=2 x d x

Name: Problem 11, Chapter 31: Basic Technical Mathematics with Calculus
Uploaded: 2021-06-05T19:12:30-08:00
Duration: 53 s
Channel: Wendi Zhao
Description: Solve the given differential equations. √(x^2+y^2) d x-2 y d y=2 x d x

step 1 First, we add 2y -D -Y to both sides. And now the right -hand side can be written as D x squared

Solve the given differential equations. √(x^2+y^2) d x-2 y d...

Key Concepts

First-Order Differential Equations

This concept deals with differential equations involving the first derivative of the unknown function. It encompasses a variety of techniques for solving equations, including separation of variables, exact equations, integrating factors, and substitution methods. Understanding these methods is crucial for identifying the most efficient path to find the solution.

Exact Differential Equations

An exact differential equation is one where a differential expression M(x, y)dx + N(x, y)dy is the total differential of some function F(x, y). The key test is verifying whether ?M/?y equals ?N/?x. Recognizing an exact equation allows one to integrate directly, reconstructing the potential function whose level sets represent the solutions.

Integrating Factors

When a differential equation is not exact, an integrating factor—a function that multiplies the entire equation—can sometimes be found to convert it into an exact equation. The integrating factor may depend on x, y, or both, and its determination is a critical technique for solving non-exact first-order differential equations.

Homogeneous Differential Equations

A differential equation is classified as homogeneous if each term is a homogeneous function of the same degree. This property allows for a substitution, such as u = y/x, to reduce the equation to one that is separable. Recognizing homogeneity helps in transforming complex differential equations into simpler forms that are easier to solve.

Change of Coordinates

Sometimes, the structure of a differential equation suggests that a coordinate transformation would simplify its form. For instance, expressions like ?(x² + y²) naturally lead to a switch to polar coordinates. This change of variables can simplify the algebra and the integration process, making it a powerful technique in differential equations.

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