consider the general real homogeneous quadratic equation u(x,y)=$$ ax^2+\text{bxy}+cy^2 $$ where a,b,c are real roots.  show that u is harmonic if and only if a=-c

To prove that the general real homogeneous quadratic equation

\[
u(x, y) = ax^2 + bxy + cy^2
\]

is **harmonic** if and only if $a = -c$, we will first recall the condition for a function to be harmonic.

### Harmonic Condition:
A twice-differentiable function $ u(x, y) $ is harmonic if it satisfies **Laplace's equation** in two variables, which is:

\[
\Delta u(x, y) = 0
\]

where $ \Delta $ is the Laplace operator in two dimensions, given by:

\[
\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}
\]

### Step 1: Compute the partial derivatives of $ u(x, y) $.

We are given:

\[
u(x, y) = ax^2 + bxy + cy^2
\]

First, we compute the second partial derivatives.

#### Compute $ \frac{\partial u}{\partial x} $:

\[
\frac{\partial u}{\partial x} = 2ax + by
\]

#### Compute $ \frac{\partial^2 u}{\partial x^2} $:

\[
\frac{\partial^2 u}{\partial x^2} = 2a
\]

#### Compute $ \frac{\partial u}{\partial y} $:

\[
\frac{\partial u}{\partial y} = bx + 2cy
\]

#### Compute $ \frac{\partial^2 u}{\partial y^2} $:

\[
\frac{\partial^2 u}{\partial y^2} = 2c
\]

### Step 2: Apply the Laplace operator.

Now we apply the Laplace operator:

\[
\Delta u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}
\]

Substituting the values of the second partial derivatives:

\[
\Delta u = 2a + 2c
\]

### Step 3: Set the Laplace equation equal to zero.

For the function $ u(x, y) $ to be harmonic, we must have:

\[
\Delta u = 0
\]

Thus,

\[
2a + 2c = 0
\]

This simplifies to:

\[
a + c = 0
\]

### Step 4: Conclusion.

Therefore, $ a = -c $ is the necessary and sufficient condition for the quadratic function $ u(x, y) = ax^2 + bxy + cy^2 $ to be harmonic.

---

Would you like more details on this, or do you have any further questions?

Here are 5 related questions you might be interested in:

1. How does Laplace's equation extend to three dimensions?
2. What is the physical significance of harmonic functions?
3. How can we solve Laplace's equation for non-quadratic functions?
4. How is the Laplace equation used in fluid dynamics or electrostatics?
5. What is the relationship between harmonic functions and potential theory?

**Tip:** Harmonic functions have the property that their value at any point is the average of their values on any surrounding circle (mean value property).

Consider the general real homogeneous quadratic equation u(x,y) = ax^2 + bxy + cy^2 where a, b, c are real roots. Show that u is harmonic if and only if a = -c.

Explore how to prove that a general real homogeneous quadratic equation u(x, y) = ax^2 + bxy + cy^2 is harmonic if and only if a = -c. This solution applies Laplace's equation and discusses key concepts in partial derivatives and harmonic functions. Ideal for undergraduate students.

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