Solving a Nonlinear Differential Equation with Variable Coefficients

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SUMMARY

The discussion centers on solving the nonlinear differential equation xy' + xy = 1 - y. The participant initially struggles with identifying the correct method to solve the equation, considering various techniques such as integrating factors and Bernoulli's formula. Ultimately, the solution involves rewriting the equation as x*y' + (x + 1)*y = 1, dividing by x, and then finding the appropriate integrating factor. This approach clarifies the solution process and highlights the importance of recognizing the equation's structure.

PREREQUISITES
  • Understanding of differential equations, particularly first-order types.
  • Familiarity with integrating factors and their application in solving differential equations.
  • Knowledge of Bernoulli's differential equation and its characteristics.
  • Ability to manipulate equations and recognize their forms for effective problem-solving.
NEXT STEPS
  • Study the method of integrating factors in detail for first-order differential equations.
  • Explore the characteristics and applications of Bernoulli's differential equations.
  • Practice rewriting and manipulating differential equations to identify solvable forms.
  • Learn about exact equations and conditions for their applicability in solving differential equations.
USEFUL FOR

Students studying differential equations, educators teaching advanced mathematics, and anyone looking to enhance their problem-solving skills in nonlinear differential equations.

iamtrojan3
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Homework Statement


this problem was on my finals and i stared at it for 30 mins straight and still didn't figure it out and now it just bothers me that i don't know how to do the first problem on my final exam.

"find the general solution to the dfq"
xy'+xy= 1-y


Homework Equations




The Attempt at a Solution


i don't think its linear cause if you do the integrating factor the right side is interms of x and y
i don't think its separable... or multiplying by 1/x and subbing v for y/x could help
i don't think its exact? or is it I am not sure
bernouli's formula won't help here.
i don't know its probably something really stupid and easy... thanks for the help!
 
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The y doesn't have to be on the right side. Write it as x*y'+(x+1)*y=1. Now divide by x and find the integrating factor.
 
OH MY GOD. there goes 20 points that i should of gotten. thank you for the helpe.
 

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