Question

(1 point) In this problem we consider an equation in differential form M d.c + N dy=0. The equation (42 +3=”y 2) dx + (422.1, + 3)dy=0 y in differential form ñ dx + Ñ dy=0 is not exact. Indeed, we have Ñ , -Ñ , For this exercise we can find an integrating factor which is a function of y alone since Ñ , - Ñ , M is a function of y alone. Namely we have (y) = Multiplying the original equation by the integrating factor we obtain a new equation M dx + N dy=0 where M= N= Which is exact since My = N = are equal. This problem is exact. Therefore an implicit general solution can be written in the form F(2,y) = C where F(x, y) =

Answer 1

Ans. The given differential equation is ( 44°+31°y-2)da +(4xy +-3 dyzo Compairing with Bu da + M dy=0 Antarzy 4m²y2 +3 o+3x²+ (-2) 9-2-1 -2 We get, Then -1 to My Then My - A 6n2-3 4x 21 x y 8 ay-t - 6x2y3 – Exy-1 -- 6x²y3 – 8xy-1 4x + 3x²4-2 -26-1 (322y2 + 4n) (4x + 3 uty-2) .24 1 M my-Na $ (4) ray. m y Clearly My - NA M is function of y only % Integrating factor is also function of y only = u(y) = - Socyndy o z dy = 42 u(y) = y² e 2logy i.e.