Question

Question 1: Let y be a function of x. Identify (do not solve) the following equations as separable, linear, exact, or having an integrating factor that is a function of either x alone or y alone. Give a clear and detailed reason for your answer (y(a) (2

Answer 1

(2y^2 + 2y + 4x^2) dx + (2xy + x) dy = 0 comparing the give equation with m dx + n dy = 0 we get , m = 2y^2 + 2y + 4x^2 7 n = 2xy + x therefore partial differential m/partial differential y = 4y + 2 and partial differential n/partial differential x = 2y + 1 therefore partial differential m/partial differential y notequal partial differential n /partial differential x , the given equation is not exact. Now , (partial differential m/partial differential y - partial differential n/partial differential x)/ n = 2y + 1/x (2y + 1) = 1/x = f (x) therefore Integrating factor = e^integral f (x) dx = e^logx = x therefore multiplying the given equation by x , it will reduce to exact differential equation and can be solved . (2y^3 + 2y^2) dx + (3y^2 x + 2xy) dy = 0 i . e y (2y^2 + 2y) dx + x (3y^2 + 2y) dy = 0 the given equation is of the form, y (x , y) dx + x f _2 (x , y) dy = 0