General solution of a differential equation

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SUMMARY

The discussion centers on solving a differential equation presented in implicit form, specifically the equation dy/dx = (cosx - sinx)e^(cosx + sinx + y). The primary method suggested for solving this equation is separation of variables, which involves manipulating the equation to isolate y on one side. A key insight provided is to express the exponential term as a product of two exponentials: exp(cosx + sinx) * exp(y). This approach facilitates the separation of variables and leads to the general solution.

PREREQUISITES
  • Understanding of differential equations, specifically first-order equations.
  • Familiarity with the separation of variables technique.
  • Knowledge of exponential functions and their properties.
  • Basic algebra skills for manipulating equations.
NEXT STEPS
  • Study the method of separation of variables in differential equations.
  • Learn about implicit solutions and their applications in differential equations.
  • Explore the properties of exponential functions in the context of differential equations.
  • Practice solving first-order differential equations using various methods.
USEFUL FOR

Students studying differential equations, educators teaching calculus, and anyone seeking to enhance their problem-solving skills in mathematical analysis.

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Homework Statement


Find the general solution to the differential equation in implicit form.
http://www.texify.com/img/%5Clarge%5C%21%5Cfrac%7Bdy%7D%7Bdx%7D%3D%28cosx-sinx%29e%5E%7Bcosx%2Bsinx%2By%7D.gif

Homework Equations


http://www.texify.com/img/%5Clarge%5C%21%5Cfrac%7Bdy%7D%7Bdx%7D%3Df%28x%29g%28y%29.gif

The Attempt at a Solution


Am I correct in assuming that this is a separation of variables problem? I can't seem to grasp the algebra to move the y to the left hand side. Could anybody give me a nudge in the right direction?
 
Last edited by a moderator:
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split exp{cosx+sinx+y} into exp(cosx+sinx)*exp(y)
 

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