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Calculus · Studypool

Calculus Question

Post ID: 3560269

  1. Question 1 — Characteristics and complete integral (20 marks)

    Consider the Lagrange first-order PDE𝑃(𝑥,𝑦,𝑧)𝑝+𝑄(𝑥,𝑦,𝑧)𝑞=𝑅(𝑥,𝑦,𝑧),where𝑝=∂𝑧∂𝑥,𝑞=∂𝑧∂𝑦.
    1. Derive the characteristic system
    𝑑𝑥𝑃=𝑑𝑦𝑄=𝑑𝑧𝑅.
    1. Explain precisely why two functionally independent first integrals
    𝑢(𝑥,𝑦,𝑧)=𝐶1,𝑣(𝑥,𝑦,𝑧)=𝐶2lead to the general integral𝐹(𝑢,𝑣)=0.
    1. State the regularity and non-characteristic assumptions required for this construction to be locally valid.

    Question 2 — Finding independent integrals (20 marks)

    Solve the PDE(𝑦+𝑧)𝑝+(𝑧+𝑥)𝑞=𝑥+𝑦.Find two independent first integrals of the characteristic system and hence obtain the general solution.Your solution must justify the functional independence of the two integrals rather than merely producing two conserved quantities.

    Question 3 — Geometric interpretation (15 marks)

    Let𝑋=𝑃(𝑥,𝑦,𝑧)∂∂𝑥+𝑄(𝑥,𝑦,𝑧)∂∂𝑦+𝑅(𝑥,𝑦,𝑧)∂∂𝑧.
    1. Interpret Lagrange's auxiliary equations in terms of the integral curves of the vector field 𝑋.
    2. Show that a first integral 𝐼(𝑥,𝑦,𝑧) satisfies
    𝑋(𝐼)=0.
    1. Explain why a solution surface 𝑧=𝑧(𝑥,𝑦) of
    𝑃𝑝+𝑄𝑞=𝑅is an integral surface of the characteristic distribution.
    1. Discuss what can fail when 𝑃=𝑄=0, or when the characteristic field becomes tangent to the projection onto the 𝑥𝑦-plane.

    Question 4 — Complete integral and singular behavior (15 marks)

    Consider𝑥𝑝+𝑦𝑞=𝑧.
    1. Construct the characteristic equations.
    2. Find two independent first integrals.
    3. Obtain the general integral.
    4. Determine whether a meaningful singular integral exists.
    5. Discuss the role of the surface 𝑥=𝑦=0 in your analysis.

    Question 5 — Cauchy problem and characteristics (15 marks)

    Consider𝑝+2𝑞=𝑥𝑧.The initial data are prescribed on the curve𝑥=0,𝑦=𝑠,𝑧=𝑠2.
    1. Set up the characteristic initial-value problem.
    2. Determine whether the initial curve is non-characteristic.
    3. Obtain the characteristic surface, at least locally.
    4. Explain the conditions under which the solution is unique.
    5. Discuss what happens if the initial curve becomes characteristic.

    Question 6 — Advanced theoretical question (15 marks)

    Let𝑃𝑝+𝑄𝑞=𝑅with 𝑃,𝑄,𝑅∈𝐶1(Ω).Suppose 𝑢,𝑣∈𝐶1(Ω) satisfy𝑃𝑢𝑥+𝑄𝑢𝑦+𝑅𝑢𝑧=0,𝑃𝑣𝑥+𝑄𝑣𝑦+𝑅𝑣𝑧=0.Prove that𝐹(𝑢,𝑣)=0defines a family of integral surfaces wherever∇𝑢×∇𝑣≠0and the projection onto the (𝑥,𝑦)-plane is locally regular.Explain carefully the distinction between:
    • a first integral,
    • a general integral,
    • a complete integral, and
    • a singular integral.