10.4. Examples of Direct Integration

Try these first

1. 1.

   Solve the PDE ∂z/∂x = x + 3. What is the function z?

   Hint

   Think about integrating with respect to x.

2. 2.

   What is an arbitrary function in the context of PDEs?

   Hint

   Remember how we treat constants in single variable integration.

3. 3.

   What method do we use to solve PDEs like ∂z/∂x = 2x + y?

   • Integration
   • Differentiation
   • Summation

   Hint

   Think about how to find the function from its rate of change.

4. 4.

   True or False: An arbitrary function of y appears during the integration process in PDEs.

   • True
   • False

   Hint

   Reconsider how constants of integration behave in multi-variable cases.

5. 5.

   Solve the PDE ∂z/∂x = 3e^x + sin(y) and indicate the form of the arbitrary function.

   Hint

   Pay close attention to the exponential function when integrating.

6. 6.

   Given the PDEs ∂z/∂x + ∂z/∂y = xy and ∂z/∂y = x-y², use both to find an expression for z.

   Hint

   Link the two equations through integration and differentiation to extract solutions.