8.4. Example Problems

Try these first

1. 1.

   Convert the PDE $\frac{\partial^2 z}{\partial x^2} - 4 \frac{\partial^2 z}{\partial y^2} = 0$ to operator form.

   Hint

   Replace the derivatives with their operator equivalents.

2. 2.

   Define what a homogeneous PDE is in your own words.

   Hint

   Think about the role of terms in the equation.

3. 3.

   What type of PDE has no free terms?

   • Homogeneous
   • Non-homogeneous
   • Linear
   • Non-linear

   Hint

   Remember the definition of homogeneous equations.

4. 4.

   True or False: A repeated root leads to a general solution involving both $f(y - mx)$ and $x f(y - mx)$.

   • True
   • False

   Hint

   Recall how the roots affect the general solution forms.

5. 5.

   Consider the PDE $\frac{\partial^2 z}{\partial x^2} + k \frac{\partial^2 z}{\partial y^2} + m \frac{\partial z}{\partial x \partial y} = 0$ where k and m are constants. Derive the general solution when k > 0 and m < 0.

   Hint

   Analyze the discriminant of the auxiliary equation to categorize the roots.

6. 6.

   Create your own PDE that is homogeneous and has a distinct root, then solve it and present your general solution.

   Hint

   Ensure your constants create a distinct quadratic equation for compliance.