A particular solution of 3 e ^x y d x+ (1- e ^x ) ^2 y dy =0 with y(1)= 4 is

Step-by-step solution

The differential equation is 3e^x y ,dx + (1 - e^x) ^2 y ,dy = 0 . Separating variables gives 3e^x 1 - e^x dx = - ^2 y y dy . Integrating both sides: 3e^x 1 - e^x dx = - ^2 y y dy . Using substitution u = 1 - e^x on the left yields -3 |1 - e^x| . With v = y on the right gives - | y| . Equating and simplifying: | y| = 3 |1 - e^x| + C , leading to y = K(1 - e^x)^3 . Applying y(1) = 4 gives 1 = K(1 - e)^3 , so K = 1 (1 - e)^3 . The particular solution is y = ( 1 - e^x 1 - e )^3 , matching option D. Final answer: D