Write the equation in the form y\' = f (y/x) then use the substitution y = xu to

Question

Write the equation in the form y' = f (y/x) then use the substitution y = xu to solve the initial value problem. Xy2y' = y3 - x3, y(2) = 2. The solution can be written in the form y= g(x) where g(x) =

Explanation / Answer

xy^2y' = y^3 - x^3 y' = (y^3 - x^3)/(xy^2) y' = y/x - (x/y)^2 y' = (y/x) - (1/(y/x))^2 => y' = f(y/x), where f(y/x) = (y/x) - (1/(y/x))^2 y = vx dy/dx = v + xdv/dx [product rule] v + xdv/dx = v - (1/v)^2 xdv/dx = -1/v^2 v^2 dv = -1/x dx integrating both sides v^3/3 = - ln x + c v = y/x => (y/x)^3 = 3ln x + c => y^3 = 3x^3ln x + cx^3 y(2) = 2 => 8 = 24ln 2 + 8c => 1 - 3ln 2 = c => y^3 = 3x^3ln x + (1 - 3ln 2) x^3 => y = cube root(3x^3ln x + (1 - 3ln 2) x^3) => y = g(x) g(x) = cube root(3x^3ln x + (1 - 3ln 2) x^3)

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