dx-(y%2Bxy)dy%3D0.jpeg "(x^2+y^2-5)dx-(y+xy)dy=0")
Solving the Differential Equation: (x^2 + y^2 - 5)dx - (y + xy)dy = 0
This article will explore the solution to the differential equation:
(x^2 + y^2 - 5)dx - (y + xy)dy = 0
This is a first-order, non-linear differential equation. We can solve this equation by utilizing the method of exact differential equations.
1. Identifying an Exact Differential Equation
A differential equation of 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) = x^2 + y^2 - 5 and N(x, y) = -(y + xy). Let's calculate the partial derivatives:
∂M/∂y = 2y ∂N/∂x = -y
Since ∂M/∂y ≠ ∂N/∂x, the given equation is not exact.
2. Finding an Integrating Factor
We can convert the given equation into an exact one by multiplying it with an integrating factor.
To find the integrating factor, we can utilize the following formula:
μ(x) = exp(∫(∂N/∂x - ∂M/∂y)/M dx)
In our case:
μ(x) = exp(∫(-y - 2y)/(x^2 + y^2 - 5) dx) = exp(∫(-3y)/(x^2 + y^2 - 5) dx)
The integral in the exponent does not have a simple solution. However, if we observe that the expression in the denominator (x^2 + y^2 - 5) is similar to the expression in M(x, y), we can try a different approach. Let's assume the integrating factor is of the form μ(y) instead of μ(x).
μ(y) = exp(∫(∂M/∂y - ∂N/∂x)/N dy) = exp(∫(2y + y)/(y + xy) dy) = exp(∫(3y)/(y + xy) dy) = exp(∫(3)/(1 + x) dy) = (1 + x)^3
Now, multiplying the original equation with this integrating factor:
(1 + x)^3 * ((x^2 + y^2 - 5)dx - (y + xy)dy) = 0
This equation is now exact since:
∂[(1 + x)^3 * (x^2 + y^2 - 5)]/∂y = 2y(1 + x)^3 ∂[-(1 + x)^3 * (y + xy)]/∂x = 2y(1 + x)^3
3. Solving the Exact Equation
Since the equation is exact, we can find a solution by integrating M(x, y) with respect to x and N(x, y) with respect to y.
∫[(1 + x)^3 * (x^2 + y^2 - 5)] dx = ∫[-(1 + x)^3 * (y + xy)] dy
After integrating, we get:
(1/4)(1 + x)^4 + (1/2)y^2(1 + x)^3 - 5(1 + x)^3 = - (1/2)y^2(1 + x)^3 + C
Simplifying and rearranging, we get the final solution:
(1/4)(1 + x)^4 + y^2(1 + x)^3 - 5(1 + x)^3 = C
Where C is the constant of integration.
Conclusion
We successfully solved the non-linear differential equation (x^2 + y^2 - 5)dx - (y + xy)dy = 0 by utilizing the method of exact differential equations. We found that the equation was not exact initially but became exact after multiplying it with the integrating factor (1 + x)^3. The final solution to the equation is (1/4)(1 + x)^4 + y^2(1 + x)^3 - 5(1 + x)^3 = C. This solution represents a family of curves in the xy-plane.