Question

Find general solutions of the differential equations to x. 14. xy ry-уз 15. y +3y 3xe3 16. y 2-2xy y2 18. 2x2y-rly,-: уз 20. xy' +3y 3x-3/2 11. x2ys xy + 3y2 25. 2y + (x +1)y'-3x +3

Answer 1

x^3 + 3y - xy' = 0 rightwardsdoublearrow xy' = x^3 + 3y rightwardsdoublearrow xy' -3y = x^3 dividing by `x' rightwardsdoublearrow y' - (-3/x) y = x^2 --------- (1) equation (1) is linear differential equation of the form y' + p(x) y = q(x), where p(x) = -3/x, q(x) = x^2 Integrating factor IF = e^integral p dx = e^integral -3/x dx rightwardsdoublearrow IF = e^-3 ln x = e^ln x^-3 = x^-3 = 1/x^3 General solution is y times IF = integral IF times q (x) dx + C y times 1/x^3 = integral 1/x^3 x^2 dx + C y/x^3 = integral 1/x dx + C = ln x + C rightwardsdoublearrow y = x^3 ln x + c x^3 xy^2 + 3y^2 - x^2 y' = 0 rightwardsdoublearrow x^2 y' = xy^2 + 3y^2 rightwardsdoublearrow x^2 dy/dx = (x + 3) y^2 separating the variables rightwardsdoublearrow 1/y^2 dy = (x + 3)/x^2 dx rightwardsdoublearrow y/y^2 dy = (1/x + 3/x^2) dx integrating