5. Equations of the form y’ = P(x)y^2 + Q(x)y + R(x) are called Riccati equations....

5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are called Riccati equations.

i) If we know a solution y = φ(x) of this equation, then any other solution

can be written in the form y(x) = φ(x)+ 1/v(x), where v(x) is an unknown

function which satisfies a certain linear equation. Using the fact that

φ and y both solve the above Riccati equation, find the differential

equation that v satisfies.

ii) Consider the equation 3y’ + y^2 +2/(x^2) = 0. Find one solution of this equation by inspection.

iii) Use the method of part(i) to find the general solution of the equation

in (ii).

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Expert Solution

Colby Messinger answered 1 year ago

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5. Equations of the form y’ = P(x)*y^2 + Q(x)*y + R(x) are called Riccati equations....

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5. Equations of the form y’ = P(x)y^2 + Q(x)y + R(x) are called Riccati equations....