(3x^2+4xy)dx+(2x^2+2y)dy=0

Jasonbradley

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Jun 16, 2024

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Solving the Differential Equation: (3x^2 + 4xy)dx + (2x^2 + 2y)dy = 0

This article will guide you through the process of solving the given differential equation:

(3x^2 + 4xy)dx + (2x^2 + 2y)dy = 0

We will utilize the concept of exact differential equations to find a solution.

Identifying an Exact Differential Equation

A differential equation of the form:

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

is considered exact if the following condition holds:

∂M/∂y = ∂N/∂x

Let's apply this to our given equation:

• M(x,y) = 3x^2 + 4xy
• N(x,y) = 2x^2 + 2y

Calculating the partial derivatives:

• ∂M/∂y = 4x
• ∂N/∂x = 4x

Since ∂M/∂y = ∂N/∂x, we confirm that the given differential equation is exact.

Finding the Solution

Now that we know the equation is exact, we can proceed to find its solution. This involves the following steps:

1. Find a function F(x,y) such that:

   • ∂F/∂x = M(x,y)
   • ∂F/∂y = N(x,y)

2. The solution to the differential equation will be given by F(x,y) = C, where C is an arbitrary constant.

Step 1:

Integrate ∂F/∂x = M(x,y) = 3x^2 + 4xy with respect to x, treating y as a constant:

F(x,y) = x^3 + 2x^2y + g(y)

Here, g(y) is an arbitrary function of y, since the integration with respect to x might have introduced terms depending only on y.

Step 2:

Now, differentiate F(x,y) with respect to y:

∂F/∂y = 2x^2 + g'(y)

We know that ∂F/∂y = N(x,y) = 2x^2 + 2y. Comparing these, we get:

g'(y) = 2y

Integrate this with respect to y to find g(y):

g(y) = y^2 + C1

Where C1 is an arbitrary constant of integration.

Step 3:

Substitute the value of g(y) back into F(x,y):

F(x,y) = x^3 + 2x^2y + y^2 + C1

Final Solution:

The solution to the given differential equation is:

x^3 + 2x^2y + y^2 = C (where C = -C1 is another arbitrary constant)

Conclusion

By recognizing the given equation as an exact differential equation and utilizing the appropriate steps, we have successfully obtained the general solution: x^3 + 2x^2y + y^2 = C. This process highlights the significance of identifying exact differentials in solving certain types of differential equations.