The differential equation dy dx P(x) Q(x)y R(x)y 2 is known as Riccatis equation (a) A Riccati equation can be solved by a succession of two substitutions provided that we know a particular solution y 1 of the equation Show that the substitution y y 1 u reduces Riccatis equation to a Bernoulli equation (4) with n 2 The Bernoulli equation can then be reduced to a linear equation by the substitution w u 1 (b) Find a one parameter family of solutions for the differential equation dy dx 4 x 2 1 x y y 2 where y 1 2yx is a known solution of the equation

Question: The differential equation dy/dx = P(x) + Q(x)y + R(x)y 2 is known as Riccatis equation. (a) A Riccati equation can be solved by a

The differential equation dy/dx = P(x) + Q(x)y + R(x)y² is known as Riccati’s equation.

(a) A Riccati equation can be solved by a succession of two substitutions provided that we know a particular solution y₁ of the equation. Show that the substitution y = y₁ + u reduces Riccati’s equation to a Bernoulli equation (4) with n = 2. The Bernoulli equation can then be reduced to a linear equation by the substitution w = u^-1.

(b) Find a one-parameter family of solutions for the differential equation

dy/dx = -4/x² – 1/x y + y²

where y₁ = 2yx is a known solution of the equation.

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