Practice Partial Differential Equations - 10 | 10. Solution of PDEs by Direct Integration | Mathematics - iii (Differential Calculus) - Vol 2
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10 - Partial Differential Equations

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Learning

Practice Questions

Test your understanding with targeted questions related to the topic.

Question 1

Easy

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 \\).

Question 2

Easy

What is an arbitrary function in the context of PDEs?

💡 Hint: Think about how constants appear in single-variable integration.

Practice 4 more questions and get performance evaluation

Interactive Quizzes

Engage in quick quizzes to reinforce what you've learned and check your comprehension.

Question 1

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.

Question 2

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

  • True
  • False

💡 Hint: Reflect on why we introduce arbitrary functions.

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Challenge Problems

Push your limits with challenges.

Question 1

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.

Question 2

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.

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