10.1. What is Direct Integration?

Try these first

1. 1.

   Explain what direct integration is in the context of PDEs.

   Hint

   Think about how we integrate in calculus.

2. 2.

   What is an arbitrary function in direct integration?

   Hint

   Consider how constants appear in regular integration.

3. 3.

   What is the first step in using direct integration?

   • Integrate with respect to one variable
   • Differentiate with respect to both variables
   • Assume values for variables

   Hint

   Think about integration as the starting point of the solution process.

4. 4.

   True or False: All PDEs can be solved using direct integration.

   • True
   • False

   Hint

   Consider the conditions we discussed for direct integration.

5. 5.

   Solve the PDE $\frac{\partial z}{\partial x} = e^x \sin(y)$ and interpret the arbitrary function in your solution.

   Hint

   Think about how integration constants behave.

6. 6.

   Given two PDEs to solve, $\frac{\partial z}{\partial x} = x^2y + 2$ and $\frac{\partial z}{\partial y} = xy + 3x$, find the complete solution.

   Hint

   Remember to use both equations to find the unique solution.