Practice Partial Differential Equations (10) - Solution of PDEs by Direct Integration

Practice: Partial Differential Equations

Lesson 10Solution of PDEs by Direct Integration

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10 questions

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6 sample questions

Question 1.
Integrate the following PDE: $\frac{\partial z}{\partial x} = 3x + y$ .
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Remember to treat \$ y \$ as a constant while integrating with respect to \$ x \$.
Question 2.
What is an arbitrary function in the context of PDEs?
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Think about how constants appear in single-variable integration.
3Question 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
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Think about how we handle one variable at a time.
4Question 4.
True or False: Arbitrary functions are constant in their respective variables.
- True
- False
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Reflect on why we introduce arbitrary functions.
Question 5.
Consider a PDE $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = e^{-x}$ . Solve the equation using direct integration.
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Look at each term and treat others as constants while integrating.
Question 6.
Given the equation $\frac{\partial z}{\partial x} = y^2 + x^2$ and simultaneously $\frac{\partial z}{\partial y} = 2xy$ , solve for $z$ .
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Each integration adds complexity layer; check dependencies between leaps in variables.

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