(3x²+2xy+4y²)dx + (x²+8xy+18y)dy=0 solve the exact differential equation

To solve the exact differential equation

\[
(3x^2 + 2xy + 4y^2) \, dx + (x^2 + 8xy + 18y) \, dy = 0,
\]

we need to first verify that the equation is exact, and then find a potential function \(F(x, y)\) such that \(dF = 0\).

### Step 1: Verify Exactness

An equation of the form \(M(x, y) \, dx + N(x, y) \, dy = 0\) is exact if

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.
\]

Here,

\[
M(x, y) = 3x^2 + 2xy + 4y^2, \quad N(x, y) = x^2 + 8xy + 18y.
\]

Compute the partial derivatives:

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (3x^2 + 2xy + 4y^2) = 2x + 8y,
\]

\[
\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (x^2 + 8xy + 18y) = 2x + 8y.
\]

Since \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the differential equation is exact.

### Step 2: Find the Potential Function \(F(x, y)\)

Since the equation is exact, there exists a function \(F(x, y)\) such that:

\[
\frac{\partial F}{\partial x} = M(x, y) = 3x^2 + 2xy + 4y^2, \quad \frac{\partial F}{\partial y} = N(x, y) = x^2 + 8xy + 18y.
\]

#### Finding \(F(x, y)\)

Integrate \(M(x, y)\) with respect to \(x\):

\[
F(x, y) = \int (3x^2 + 2xy + 4y^2) \, dx = x^3 + x^2y + 4xy^2 + g(y),
\]

where \(g(y)\) is a function of \(y\) alone.

Next, differentiate \(F(x, y)\) with respect to \(y\) and set it equal to \(N(x, y)\):

\[
\frac{\partial F}{\partial y} = \frac{\partial}{\partial y} \left( x^3 + x^2y + 4xy^2 + g(y) \right) = x^2 + 8xy + g'(y).
\]

Set this equal to \(N(x, y)\):

\[
x^2 + 8xy + g'(y) = x^2 + 8xy + 18y.
\]

Simplify the equation:

\[
g'(y) = 18y.
\]

Integrate \(g'(y)\) with respect to \(y\):

\[
g(y) = \int 18y \, dy = 9y^2 + C.
\]

Thus, the potential function \(F(x, y)\) is:

\[
F(x, y) = x^3 + x^2y + 4xy^2 + 9y^2 + C.
\]

### Step 3: Solve for the Implicit Solution

The general solution is given by \(F(x, y) = C_1\). Set \(F(x, y)\) equal to a constant \(C_1\):

\[
x^3 + x^2y + 4xy^2 + 9y^2 = C_1.
\]

This is the implicit solution of the given exact differential equation.

### Conclusion

The solution to the differential equation is:

\[
\boxed{x^3 + x^2y + 4xy^2 + 9y^2 = C_1.}
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you determine if a differential equation is exact?
2. What methods can be used to solve non-exact differential equations?
3. How can integrating factors be used to make a differential equation exact?
4. What is the significance of the potential function in solving exact differential equations?
5. How does an initial condition affect the solution of an exact differential equation?

### Tip:

When solving exact differential equations, always check the partial derivatives of the given functions to verify exactness. If the equation is not exact, consider finding an integrating factor to make it exact.

Learn how to solve the exact differential equation (3x² + 2xy + 4y²)dx + (x² + 8xy + 18y)dy = 0 with a step-by-step solution. Verify exactness, find the potential function, and derive the implicit solution. Suitable for advanced undergraduate students and enthusiasts of differential equations.

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