Solving a Nonlinear Differential Equation with Variable Coefficients

In summary, the problem is to find the general solution to the differential equation xy'+xy=1-y. The student struggled for 30 minutes and tried various methods such as integrating factor and substituting v for y/x, but could not find the solution. They also ruled out other methods such as separable and Bernoulli's formula. Finally, they received help and realized that dividing by x and finding the integrating factor would solve the problem. They express gratitude for the help and regret not realizing it sooner.
  • #1
iamtrojan3
56
0

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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  • #2
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.
 
  • #3
OH MY GOD. there goes 20 points that i should of gotten. thank you for the helpe.
 

1. What is a differential equation?

A differential equation is a mathematical equation that involves an unknown function and its derivatives. It is used to describe the relationship between a quantity and its rate of change.

2. Why are differential equations important?

Differential equations are important because they are used to model and solve real-world problems in various fields such as physics, engineering, economics, and biology. They provide a powerful tool for understanding and predicting how systems change over time.

3. What are the different types of differential equations?

The main types of differential equations are ordinary differential equations (ODEs), which involve a single independent variable, and partial differential equations (PDEs), which involve multiple independent variables. ODEs can be further classified as linear or nonlinear, and PDEs can be classified as elliptic, hyperbolic, or parabolic.

4. How do I solve a differential equation?

Solving a differential equation involves finding a function that satisfies the equation. This can be done analytically, by using mathematical techniques such as separation of variables, or numerically, by using computational methods such as Euler's method or Runge-Kutta methods.

5. What skills are needed to understand and solve differential equations?

To understand and solve differential equations, one needs a strong foundation in calculus, particularly in differentiation and integration. Knowledge of linear algebra and differential equations theory is also helpful. Additionally, problem-solving skills and the ability to think critically and creatively are important for successfully solving differential equations.

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