Show that the differential equation (1 + y^2)dx + (1 - x)dy = 0 is exact and find the integrating factor by inspection and solve it.
Understand the Problem
The question is asking to show that a differential equation is exact and to find the integrating factor, followed by solving the equation. It involves understanding differential equations and their properties.
Answer
The solution is given by $$ x + \ln(1 + y^2) = y + C $$
Answer for screen readers
The solution to the differential equation is $$ x + \ln(1 + y^2) = y + C $$
Steps to Solve
- Identify the Differential Equation
We begin with the differential equation given by: $$ (1 + y^2) \frac{dx}{dy} + (1 - x) dy = 0 $$
- Rearrange the Equation
Rearranging the equation helps in identifying the function forms: $$ (1 + y^2) \frac{dx}{dy} = -(1 - x) $$ This can be rewritten as: $$ \frac{dx}{dy} = -\frac{(1-x)}{(1+y^2)} $$
- Check if the Equation is Exact
An equation is exact if: $$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} $$ where (M = 1 + y^2) and (N = -(1-x)).
Calculate: $$ \frac{\partial M}{\partial y} = 2y $$ and $$ \frac{\partial N}{\partial x} = 1 $$
Since these are not equal, the equation is not exact.
- Finding an Integrating Factor
An integrating factor typically depends on (x) or (y). Here we try ( \mu(y) = \frac{1}{1 + y^2} ).
Multiply the entire equation by ( \mu(y) ): $$ \frac{1}{1+y^2} \left((1+y^2)\frac{dx}{dy} + (1-x) dy\right) = 0 $$ This simplifies to: $$ \frac{dx}{dy} + \frac{(1-x)}{(1+y^2)} dy = 0 $$ This new equation may be checked for exactness again.
- Solve the Exact Equation
Now, integrating: $$ \int M , dy = \int (1) , dy = y + C $$ Next, solve: $$ \frac{dx}{dy} = -\frac{(1-x)}{(1+y^2)} \rightarrow \text{Integrate both sides. } $$
The solution to the differential equation is $$ x + \ln(1 + y^2) = y + C $$
More Information
This differential equation is a first-order, and we used the concept of exactness and integrating factors to find the solution. The integrating factor allowed us to solve it as an exact differential equation, simplifying the integration process.
Tips
- Neglecting to check for exactness before proceeding with solving can lead to confusion.
- Misidentifying the correct integrating factor or using a wrong approach to find it can complicate the solving process.