Week 6

Chapters:

• Differential Equations For Engineers

  • 2.5 Equations Solvable for the Independent or Dependent Variable
    • Case 1. Equation Solvable for Variable y
    • Case 2. Equation Solvable for Variable x
    • The Clairaut Equation

• Handbook of Differential Equations

  • 50 Clairaut’s Equation
  • 81 Lagrange's Equation

• Ordinary Differential Equations 📜

  • 2.44 Equations linear in x and y
  • 2.45 The Clairaut Equation

Videos:

• Mike, the Mathematician: The Clairaut Differential Equation and Singular Solutions

Lagrange's Equation

If your lovely teacher said Lagrange without further explanation, she means the ordinary form of Lagrange's Equation, which may be called d'Alembert's Equation:

• Wolfram: Lagrange's Equation
• Wolfram: d'Alembert's Equation
• Quoting Ordinary Differential Equations, 2.44. Equations linear in x and y, p. 38:

  The equations appear to have been integrated by John Bernoulli before the year 1694. Its singular solutions were studied by d'Alembert, Hist. Acad. Berlin 4 (1748), p. 275.

• Quoting Handbook of Differential Equations, 81. Lagrange's Equation, p. 365:

  Equation (81.1) is known as d’Alembert’s equation and also as an equation linear in x and y.

First Order Non-Linear / First Order Higher Degree

If your lovely teacher was solving equations about this, she may without mentioning solve an equation that is a particular case of Lagrange's / d'Alembert's equation called The Clairaut equation:

• Wolfram: Clairaut's Differential Equation
• Wikipedia: Clairaut's equation

Do NOT be confused with Euler-Lagrange Differential Equation:

• Wolfram: Euler-Lagrange Differential Equation