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  3. 5. equations of the form y’ = p(x)*y^2 + q(

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

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Basic Math

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

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

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

function which satisfies a certain linear equation. Using thefact that

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

equation that v satisfies.

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

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

in (ii).

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