10. Partial Differential Equations

Try these first

1. 1.

   Integrate the following PDE: $\frac{\partial z}{\partial x} = 3x + y$.

   Hint

   Remember to treat \\( y \\) as a constant while integrating with respect to \\( x \\).

2. 2.

   What is an arbitrary function in the context of PDEs?

   Hint

   Think about how constants appear in single-variable integration.

3. 3.

   What does Direct Integration allow us to do when solving PDEs?

   • Integrate multiple times
   • Solve any PDE
   • Integrate step-by-step
   • Only solve first-order PDEs

   Hint

   Think about how we handle one variable at a time.

4. 4.

   True or False: Arbitrary functions are constant in their respective variables.

   • True
   • False

   Hint

   Reflect on why we introduce arbitrary functions.

5. 5.

   Consider a PDE $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = e^{-x}$. Solve the equation using direct integration.

   Hint

   Look at each term and treat others as constants while integrating.

6. 6.

   Given the equation $\frac{\partial z}{\partial x} = y^2 + x^2$ and simultaneously $\frac{\partial z}{\partial y} = 2xy$, solve for $z$.

   Hint

   Each integration adds complexity layer; check dependencies between leaps in variables.