(1+x+xy^2)dy+(y+y^3)dx=0

5 min read Jun 16, 2024

Solving the Differential Equation: (1+x+xy^2)dy + (y+y^3)dx = 0

This article explores the solution of the given differential equation:

(1+x+xy^2)dy + (y+y^3)dx = 0

This equation is classified as a first-order, non-linear differential equation. To solve it, we will employ the method of exact differential equations.

Identifying an Exact Differential Equation

A differential equation in the form:

M(x,y)dx + N(x,y)dy = 0

is considered exact if:

∂M/∂y = ∂N/∂x

In our case:

M(x,y) = y + y^3
N(x,y) = 1 + x + xy^2

Let's calculate the partial derivatives:

∂M/∂y = 1 + 3y^2
∂N/∂x = 1 + y^2

Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact in its current form.

Finding an Integrating Factor

To make the equation exact, we need to find an integrating factor, denoted by μ(x,y). This factor will multiply the entire equation, making it exact.

We can use the following formula for finding the integrating factor:

μ(x,y) = exp(∫(∂N/∂x - ∂M/∂y)/M dx)

Let's apply this to our equation:

μ(x,y) = exp(∫(1 + y^2 - (1 + 3y^2))/(y + y^3) dx)

Simplifying:

μ(x,y) = exp(∫-2y^2/(y + y^3) dx)

μ(x,y) = exp(-2∫(1/y - 1/(1+y^2)) dx)

Integrating:

μ(x,y) = exp(-2(ln|y| - arctan(y)))

μ(x,y) = exp(-2ln|y| + 2arctan(y))

μ(x,y) = exp(2arctan(y)) / y^2

Now, we multiply the original equation by this integrating factor:

(exp(2arctan(y)) / y^2)(1+x+xy^2)dy + (exp(2arctan(y)) / y^2)(y+y^3)dx = 0

This equation is now exact because:

∂/∂y [(exp(2arctan(y)) / y^2)(y + y^3)] = ∂/∂x [(exp(2arctan(y)) / y^2)(1 + x + xy^2)]

Solving the Exact Equation

Since the equation is now exact, we can find a solution by integrating M(x,y) with respect to x, treating y as a constant. The result should be a function F(x,y) that satisfies:

dF/dx = M(x,y) dF/dy = N(x,y)

Let's find F(x,y):

F(x,y) = ∫[(exp(2arctan(y)) / y^2)(y+y^3)] dx

F(x,y) = (exp(2arctan(y)) / y^2) [xy + xy^3/3] + g(y)

where g(y) is an arbitrary function of y.

To find g(y), we differentiate F(x,y) with respect to y and equate it to N(x,y):

∂F/∂y = (exp(2arctan(y)) / y^2) [x + xy^2] + g'(y) = (exp(2arctan(y)) / y^2)(1 + x + xy^2)

Therefore, g'(y) = 0 and g(y) = C, where C is a constant.

The solution to the differential equation is given by:

F(x,y) = (exp(2arctan(y)) / y^2) [xy + xy^3/3] + C = 0

This implicit equation represents the general solution to the original differential equation.

Conclusion

We successfully solved the given differential equation by first identifying that it was not exact. We then found an integrating factor that made the equation exact. Finally, we integrated the exact equation to obtain the general solution.

Remember that this solution is an implicit equation, and further analysis might be necessary to obtain an explicit solution, if possible.