ECUACIÓN DIFERENCIAL HOMOGÉNEA | Cambio y = xv paso a paso

Item: ECUACIÓN DIFERENCIAL HOMOGÉNEA | Cambio y = xv paso a paso
Rating: 4
Author: PostSingularity Institute

HOMOGENEOUS DIFFERENTIAL EQUATION | Substitution y = xv step by step

by Matemáticas con Juan

Formal & Physical Sciences Mathematics PBKCalculus and mathematical analysis PBKJDifferential calculus and equations

🎙 Matemáticas con Juan 👥 2.1M 📅 September 7, 2026 ⏱ 14 min 👁 82 📄 tutorial 🧭 2026-09-07

Available in: English (current) Français

Keywords

homogeneous differential equationy=xv substitutionseparable variablesintegrationarctangent

Summary

The video presents a detailed, step-by-step solution of the homogeneous differential equation (x+y)dx + (y-x)dy = 0. The instructor first explains the general definition of a homogeneous differential equation, verifying that the functions M(x,y) and N(x,y) are homogeneous of degree 1. Then, the substitution y = xv and dy = v dx + x dv is introduced, transforming the equation into a separable form. After separating variables, both sides are integrated, yielding arctangent and logarithmic terms. The instructor then reverts to the original variables, using properties of logarithms to simplify the expression. The final implicit solution is arctan(y/x) - (1/2)ln(x² + y²) + C = 0. The video concludes with references to additional resources, including a playlist of homogeneous differential equations and a mega-class covering 50 differential equations.

128 words

Critical Evaluation

Value of the Information & Strength of the Argument

The video provides a clear and methodical demonstration of solving a homogeneous differential equation. The argumentation is solid: the instructor verifies the homogeneity condition, applies the substitution correctly, and carefully performs algebraic manipulations to separate variables and integrate. The steps are logically sequenced, and the final solution is correctly derived. The value lies in its pedagogical clarity, making the method accessible to learners, though it does not offer novel insights beyond standard textbook techniques.

Scientific Rigor, Source Quality, Title Accuracy

The mathematical rigor is high; the solution is correct and well-explained. The video does not cite external sources, but it references its own playlist and a mega-class video in the description, which are relevant for further practice. The title accurately reflects the content, and the video’s structure with chapters aids comprehension. No comments were provided for analysis.

146 words

Title / Content Match

The title accurately describes the content: a step-by-step solution of a homogeneous differential equation using the substitution y=xv.

Quality & Reliability

8/10

The video presents a rigorous, step-by-step solution of a homogeneous differential equation, correctly applying the substitution y=xv and verifying homogeneity conditions. The mathematical steps are accurate and clearly explained, though the presentation is informal and lacks formal citations.

Chapters

Cited Sources

Playlist: Ecuaciones diferenciales homogéneas — Referenced at the end of the video as a collection of more homogeneous differential equation exercises.
Mega clase: 50 ECUACIONES DIFERENCIALES — Referenced as a comprehensive class covering 50 differential equations.

Concurring Sources

Homogeneous differential equation - Wikipedia — Confirms the definition and substitution method used in the video.

Contribution & Novelties

The video offers a clear, step-by-step tutorial on solving homogeneous differential equations, reinforcing standard methods. Its originality lies in the pedagogical approach, breaking down each step and explaining the reasoning. For further exploration, consider the following:

Pour aller plus loin :

Homogeneous differential equation (Wikipedia) — Provides a formal definition and methods.
Separation of variables (Wikipedia) — Explains the technique used after substitution.
List of integrals of rational functions — Useful for understanding the integration steps.

76 words

Radar Profile

The radar profile shows high scores in quality of information and reliability, with moderate scores in quantity and technical level. This indicates a focused, accurate tutorial that provides sufficient detail for understanding, though it does not cover a broad range of topics.

Reliability 8/10