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Equations of Bernoulli, Ricatti, and Clairaut: Solving Methods

Learn how to solve Bernoulli’s, Ricatti’s, and Clairaut’s equations by applying specific substitution techniques and methods. Understand the principles behind each equation and solve them step by step. Explore parametric solutions for Clairaut’s equation.

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Equations of Bernoulli, Ricatti, and Clairaut: Solving Methods

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  1. Section 2.6 Equations of Bernoulli, Ricatti, and Clairaut

  2. BERNOULLI’S EQUATION The differential equation where n is any real number, is called Bernoulli’s equation. For n = 0 and n = 1, the equation is linear and can be solved by methods of the last section.

  3. SOLVING A BERNOULLI EQUATION 1. Rewrite the equation as 2. Use the substitution w = y1 − n, n ≠ 0, n ≠ 1. NOTE: dw/dx = (1 − n) y−ndy/dx. 3. This substitution turns the equation in Step 1 into a linear equation which can be solved by the method of the last section.

  4. RICATTI’S EQUATION The nonlinear differential equation is called Ricatti’s equation.

  5. SOLVING A RICATTI EQUATION 1. Find a particular solution y1. (This may be given.) 2. Use the substitution y = y1 + u and to reduce the equation to a Bernoulli equation with n = 2. 3. Solve the Bernoulli equation.

  6. CLAIRAUT’S EQUATION The nonlinear differential equation y = xy′ + f (y′) is called Clauraut’s equation. Its solution is the family of straight lines y = cx + f (c), where c is an arbitrary constant. (See Problem 29.)

  7. PARAMETRIC SOLUTION TO CLAIRAUT’S EQUATION Clairaut’s equation may also have a solution in parametric form: x = −f ′ (t), y = f (t) − tf ′ (t). NOTE: This solution will be a singular solution since, if f″(t) ≠ 0, it cannot be determined from the family of solutions y = cx + f (c).

  8. HOMEWORK 1–23 odd, 27, 29, 30

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