-dx-(x%5E2%2Bxy)dy%3D0.jpeg "(3xy+y^2) Dx (x^2+xy)dy=0")
Exact Differential Equations: Solving (3xy+y^2) dx + (x^2+xy)dy = 0
In this article, we will discuss the solution of the exact differential equation (3xy+y^2) dx + (x^2+xy)dy = 0. This type of equation is commonly encountered in physics, engineering, and other fields where differential equations are used to model real-world phenomena.
Introduction to Exact Differential Equations
A differential equation is said to be exact if it can be written in the form:
M(x,y) dx + N(x,y) dy = 0
where M(x,y) and N(x,y) are functions of x and y, and:
(∂M/∂y) = (∂N/∂x)
This equation is exact if the partial derivatives of M with respect to y and N with respect to x are equal. In this case, we can find a function F(x,y) such that:
dF = M dx + N dy
F(x,y) is called the potential function, and it satisfies the equation:
∂F/∂x = M and ∂F/∂y = N
Solving the Given Equation
Now, let's solve the given equation:
(3xy+y^2) dx + (x^2+xy)dy = 0
First, we need to check if the equation is exact. We can do this by checking if:
(∂M/∂y) = (∂N/∂x)
In this case, M(x,y) = 3xy+y^2 and N(x,y) = x^2+xy. We can compute the partial derivatives:
∂M/∂y = 3x + 2y ∂N/∂x = 2x + y
Fortunately, (∂M/∂y) = (∂N/∂x), so the equation is exact.
Finding the Potential Function
Since the equation is exact, we can find a potential function F(x,y) such that:
∂F/∂x = M and ∂F/∂y = N
We can start by integrating M with respect to x:
F(x,y) = ∫(3xy+y^2) dx = x^2y + xy^2 + φ(y)
where φ(y) is an arbitrary function of y.
Next, we can differentiate F with respect to y and set it equal to N:
∂F/∂y = x^2 + xy + φ'(y) = x^2 + xy
Since φ'(y) = 0, we have φ(y) = C, where C is a constant.
Therefore, the potential function F(x,y) is:
F(x,y) = x^2y + xy^2 + C
General Solution
The general solution to the differential equation is:
F(x,y) = x^2y + xy^2 = C
where C is an arbitrary constant.
This equation describes a family of curves in the xy-plane. Each curve corresponds to a specific value of C.
Conclusion
In this article, we have solved the exact differential equation (3xy+y^2) dx + (x^2+xy)dy = 0 using the method of exact differential equations. We have found the potential function F(x,y) and the general solution to the equation. This solution can be used to model real-world phenomena in physics, engineering, and other fields.
