If the lines represented by the equation 2x^2 3xy y^2 = 0 make angles α and β with xaxis, then cot^2 α cot^2 βConsider 2x^ {2}3xy2y^ {2}2x11y12 as a polynomial over variable x Find one factor of the form kx^ {m}n, where kx^ {m} divides the monomial with the highest power 2x^ {2} and n divides the constant factor 2y^ {2}11y12 One such factor is 2xyMultiplying both sides by x one obtain the equation M (x,y)dx N (x,y)dy =0, with M (x,y) = 3x^2y xy^2 , N/x,y) = x^3 x^2y Now we have an exact equation because dM/dy = dN/dx = 3x^2 2xy We know that exist the general integral F (x,y) = C such that (1) dF/dx = M = 3x^2y xy^2 (2) dF/dy = N = x^3 x^2y
Misc 4 Prove X2 Y2 C X2 Y2 2 Is General Solution Of
If u(x y)=x^2 y^2 2x-3xy then