Expressing a differential equation into a different format

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SUMMARY

The discussion focuses on transforming the differential equation (dy/dx) = (y/x) + tan(y/x) into the form Mdx + Ndy = 0, where M and N are functions of x and y. The proposed solution suggests setting M = (y/x) + tan(y/x) and N = -1. This approach effectively reconfigures the equation, allowing for further analysis and solution techniques applicable to first-order differential equations.

PREREQUISITES
  • Understanding of differential equations, specifically first-order equations.
  • Familiarity with the concept of separable equations.
  • Knowledge of trigonometric functions and their properties.
  • Basic skills in algebraic manipulation of equations.
NEXT STEPS
  • Research methods for solving first-order differential equations.
  • Explore the implications of expressing equations in the Mdx + Ndy = 0 format.
  • Learn about the applications of trigonometric functions in differential equations.
  • Study the concept of exact equations and integrating factors.
USEFUL FOR

Mathematics students, educators, and professionals working with differential equations, particularly those interested in analytical methods for solving first-order equations.

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How do we express this differential equation (dy/dx)= (y/x) + tan(y/x) into this form( Mdx + Ndy=0) where M,N are functions of (x,y) ?
 
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\frac{dy}{dx}=\frac{y}{x}+\tan\frac{y}{x}=-\frac{M}{N}
why do not you make
M=\frac{y}{x}+\tan \frac{y}{x},N=-1
 
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