Practice Solved Examples (5.5) - Lagrange’s Linear Equation - Mathematics - iii (Differential Calculus)

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6 sample questions

Question 1.
Identify P, Q, and R in the equation ∂z/∂x + 2∂z/∂y = 3z.
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Express the PDE in the standard form Pp + Qq = R.
Question 2.
What is the general form of a first-order PDE?
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Recall the elements of the equation.
3Question 3.
What is the form of Lagrange's Linear Equation?
- Pp + Qq = 0
- Pp + Qq = R
- ∂z/∂x + ∂z/∂y = z
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Remember the equation form we discussed in class.
4Question 4.
Auxiliary equations arise from what kind of transformations?
- True
- False
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Think about how we derived them in examples.
Question 5.
Given the PDE ∂z/∂x + ∂z/∂y = 5, use Lagrange's method to find the general solution and explain the steps.
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Pay attention to the integration steps and constants involved.
Question 6.
For the equation ∂z/∂x + (sqrt(x) + 1)∂z/∂y = 0, derive the general solution using Lagrange's characteristics.
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Notice how sqrt(x) influences the auxiliary equation structure.

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