How Do You Solve (2x-y+1)dx+(x+y)dy=0 Using a Linear Transformation?

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SUMMARY

The discussion focuses on solving the differential equation (2x-y+1)dx+(x+y)dy=0 through linear transformations. Participants emphasize the importance of finding the intersection of the lines y=2x+1 and y=-x to simplify the equation by translating the axes. The transformation involves substituting x=u+h and y=v+k to eliminate constant terms, resulting in a homogeneous equation. The conversation also touches on the differences between linear and affine transformations, with participants exploring various methods to analyze and plot the solutions.

PREREQUISITES
  • Understanding of first-order ordinary differential equations (ODEs)
  • Familiarity with linear and affine transformations
  • Knowledge of homogeneous equations and their properties
  • Basic skills in plotting mathematical functions and using software like Mathematica
NEXT STEPS
  • Study the method of linear transformations in solving ODEs
  • Learn about homogeneous equations and their applications in differential equations
  • Explore the differences between linear and affine transformations in depth
  • Investigate numerical methods for solving and plotting ODEs using Mathematica
USEFUL FOR

Mathematicians, students studying differential equations, and anyone interested in advanced mathematical modeling techniques.

  • #31
Yes, you're right~
There's lots of different approaches to a question~
It's just that...
You guys are amazing!
I'd never of thought to come up with those ideas~
It makes me want to start jumping into mathematics books and brush up on my math~
 
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  • #32
After homogenizing, the generic solution to the polar form looks like the product of these four terms to various powers: (if I haven't made a mistake)

(A + B tan θ + C tan² θ)
exp(arctan(D tan θ + E))
sec θ
exp(θ)

times some constant.

Sigh, not all that enlightening. :frown: Everything except the "some constant" depend on the coefficients in the original problem.
 

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