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Exact Differential Equations: Solving (3x^2+4xy)dx+(2x^2+2y)dy=0
In this article, we will explore the concept of exact differential equations and solve the equation (3x^2+4xy)dx+(2x^2+2y)dy=0 using the method of exact differential equations.
What are Exact Differential Equations?
A differential equation of the form:
M(x,y)dx + N(x,y)dy = 0
is called an exact differential equation if there exists a function F(x,y) such that:
∂F/∂x = M(x,y) and ∂F/∂y = N(x,y)
In other words, an exact differential equation is a differential equation that can be written in the form of a total differential of a function F(x,y).
Solving the Equation (3x^2+4xy)dx+(2x^2+2y)dy=0
Now, let's solve the given equation:
(3x^2+4xy)dx + (2x^2+2y)dy = 0
To determine if this equation is exact, we need to check if the following condition is satisfied:
∂/∂y (3x^2+4xy) = ∂/∂x (2x^2+2y)
Computing the partial derivatives, we get:
∂/∂y (3x^2+4xy) = 4x ∂/∂x (2x^2+2y) = 4x
Since the condition is satisfied, the equation is exact.
Finding the Function F(x,y)
To find the function F(x,y), we can integrate M(x,y) with respect to x and N(x,y) with respect to y:
F(x,y) = ∫(3x^2+4xy)dx + ∫(2x^2+2y)dy
Evaluating the integrals, we get:
F(x,y) = x^3 + 2x^2y + x^2 + y^2 + C
where C is the constant of integration.
General Solution
The general solution to the exact differential equation is:
x^3 + 2x^2y + x^2 + y^2 = C
where C is an arbitrary constant.
Conclusion
In this article, we have solved the exact differential equation (3x^2+4xy)dx+(2x^2+2y)dy=0 using the method of exact differential equations. We have found the function F(x,y) and obtained the general solution to the equation.
