Question

explain please

2. Which one of the following DE is exact? a. (x+y)dx+(xy+1) dy=0 b (e + y)<x+ſe+x)dy = 0 c.(ye* +1) dx +(e' + xy) dy = 0 d. (sin x+cos y) dx +(cos x +sin y) dy = 0 e. (eº+1) dx +(e? + 2) dy = 0 3. The solution of the following separable DE xy' =-y? is a. y= '+c b. y=- c. y = In x+c In x+c d. In y=x? + e. yer+C 4. Consider the linear DE X'y'= x + xy? Which value of k, the given De will be homogenous? a. k=1 b. k=2 c. k=3 d. k=4 e. none of t 5. A suitable form of the general solution to the y' - 2y + y = x? +l by the undetermine coefficient method is a. ce" +C,xe" + Axe' + Bx +C b. ce" +c,x^ + Axe" + Bx +C C.c,e' + Axe' + Bx? +C d. exe" +,e' + 4x + B e. ce' + cxe+ Ax? + Bx +C

Answer 1

- Date: is is 1 of the form exact it Given Differential ton - M da + N dy o om - ƏN oy ax. We check I a) (x+y) dat(xy+1) dy=0 M=x+y, N=xyt OM = 1, 2N = y OM I ON by ax Hence @ is not exact. I b) Cexty) dat (el + a) dy = 0 m=eaty N=eY+x OM 1 ON = 1 ох = am = ON 04 03 Hence 6 is Exact D.E. e (ye2+1) dat ledt ay) dy so OM = yex ON = (x+y 22 om tan ay ax. Hence ① is not Exact d (Sina tcosyl dx + (cosatsiny) dy-o DM - coss ON = - sina azy - siny az

sy aa Hence am # ON Hence dy is not Exact. ex (e?y t) dx + Cexy + 2) dy=0 am = eny (00 Bay ON = exy (4) ax am tan ay ax Hence e not exact- 3 - 0 1 xyl= -42 x dy = -42 da ady = -42 da dy =-da y2 x Integrating We get - - 1n e tc y . = tox-c! L = 1nat c. yinate in arc Hence Solution of Dis o is y y . an xtc

4) Linear DE xtyl = 23 +9y2 ak dy = x3 + xy2 dy = dx x3 + x y z then k=3 given Linear DE is homogeneous dy = 23+2'42 2'12 da x² 1+2=3 23 5 y'-24' + y = x ² + 1 - D Given Deis non bomogeneous Associated Homogeneous Equation is 1 -2 y + y = 0 To find general solution of D m2 - 2m +10 (m-1)2 =0 M=1 y = cie?+ (2 Xea is general Solution of Now We have to find particular Solution of o is Yo = G CA+BX+ A 22 - put in o Because We get non homo geneous team is a²+1 three times m=0,0,0 then solution is el

Sutaible form of general solution of o is y = yg the у сеСо го 4 Аа? В с