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Q2. The differential equation +p(x)y+q(x)y² = r(x) is known as Riccati's equation. If one dy dæ particular solution, say y1 (x), of the Riccati's equation is known, then its solution y = y(x) can be obtained by two consecutive substitutions. (i). Show that the substitution y = y1 + u reduces Riccati's equation to a Bernoulli equation, and specify n. (ii). Transform the Bernoulli equation obtained in (i) to linear equation.

Q2. The differential equation +p(x)y+q(x)y² = r(x) is known as Riccati's equation. If one dy dæ particular solution, say y1 (x), of the Riccati's equation is known, then its solution y = y(x) can be obtained by two consecutive substitutions. (i). Show that the substitution y = y1 + u reduces Riccati's equation to a Bernoulli equation, and specify n. (ii). Transform the Bernoulli equation obtained in (i) to linear equation.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 1CR
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Q2. The differential equation + p(x)y+ q(x)y² = r(x) is known as Riccati's equation. If one dy dx particular solution, say y1 (x), of the Riccati's equation is known, then its solution y = y(x) can be obtained by two consecutive substitutions. (i). Show that the substitution y = y1 + u reduces Riccati's equation to a Bernoulli equation, and specify n. (ii). Transform the Bernoulli equation obtained in (i) to linear equation.
Transcribed Image Text:Q2. The differential equation + p(x)y+ q(x)y² = r(x) is known as Riccati's equation. If one dy dx particular solution, say y1 (x), of the Riccati's equation is known, then its solution y = y(x) can be obtained by two consecutive substitutions. (i). Show that the substitution y = y1 + u reduces Riccati's equation to a Bernoulli equation, and specify n. (ii). Transform the Bernoulli equation obtained in (i) to linear equation.
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