How do you solve (secx)(dy/dx) = e^(y + sinx)?

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SUMMARY

The differential equation (secx)(dy/dx) = e^(y + sinx) can be solved by recognizing its separable form. By rewriting the equation as e^(-y)dy = cos(x)e^(sin(x))dx, integration becomes straightforward. The solution involves integrating both sides, leading to a solution that can be expressed in terms of y and x. This method effectively utilizes the properties of exponential functions and trigonometric identities.

PREREQUISITES
  • Understanding of differential equations, specifically separable equations.
  • Familiarity with integration techniques for exponential and trigonometric functions.
  • Knowledge of the secant function and its relationship to cosine.
  • Ability to manipulate logarithmic and exponential expressions.
NEXT STEPS
  • Study techniques for solving separable differential equations.
  • Learn integration methods for functions involving e^(sin(x)).
  • Explore the properties of the secant function and its derivatives.
  • Practice solving similar differential equations to reinforce understanding.
USEFUL FOR

Students studying calculus, particularly those focusing on differential equations, as well as educators looking for examples of separable equations in teaching materials.

dmitriylm
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Homework Statement


(secx)(dy/dx) = e^(y + sinx)


Homework Equations


None


The Attempt at a Solution


Not sure. I know I can do ln of both sides and isolate the y+sinx but then I get stuck with log^(secx)
 
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Ok, I got to (e^-y)dy = (e^sinx)(cosx)dx so I should be ok from there. Just posting in case someone is interested in the future.
 
Right. Writing the equation as sec(x) dy/dx= e^y e^{sin(x)} show that it is separable. e^{-y}dy= cos(x)e^{sin(x)}dx and both of those are easy to integrate.
 

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