Question

could you answer all questions

14. Solve the initial-value problem for the exact ordinary differential equation. 2ry-9+(2y + 2 +1)y-0, y (0) --3 OAy- r O B. y O C. y O D. y O E. y OF.y O G. y rT VZS122E25 15. Solve the initial-value problem for the exact ordinary differential equation. 2ry-9r2 +(2y + x2+1)y-0, v(0) 2. -1+V12r2125 21 OA.y t B. y nren OC.y= e OD. y ra OE. y lag OFy= G. y ipprypr 1+v 12 2r2: 25 -2-1+Vn 2 16. Solve the initial-value problem associated with the exact ODE (3 y ey- 1)(2ye3sy + 3r ye)0 y(0)=1 { OA. yPe3y 1. O B. y e3y +z° = 1. O C. yPe3y- sin(z) = 1. O D. y ey-In(z + 1) = 1. O E. yPey + 1n(z + 1) = 1. OF.y e3ryr 1. G. y ery+ = 1

Answer 1

Given DE 214–922 +(24+2+1) y'=0, 46=3 compare @ with Mt Nylo, then M= 214-937, N= 247-22+1 Calculate My (289-932), Ny (24+*+1) = 22(1)-0 =0+23+0 My = 20 1 Ngc = 2x. - My ='N is Gues DE is escact. It's solution is andat (Germs of N free from alay 1. Treat gas constant s zay-2022) eta + $(2341) dy=c g/2000- Saada + Szyddst lady=c 92017 - 29.2 3+2 4 1 1 g =c Grz 3x3 +974=c -- but 9(0)=-3 ie y=-3 to alzo put these values in , we get -3) (02) -303)+ G31763)=C 0--079-320 6=0 put c=6 in , we get

yaz_ 3x375 fy=6. y²+ (2²+1)% -303_6=0. sy= -(0&11) I Cazie uro. 234= -2²-1 201) I J at 2012+1+1223+24 => y= 2 -x-1 t/a41223+298725 2 Thus 9(0)=-0²-17 Votot 9602 TBS 40s= - Itsar yoz --ves 960)= 2,. 40= -3. we have yos=-3, which is to y=-2²1-/2"+12284282725 in solution is ly=-x²1-√x+12x²+224 25 2 correct option is 'E'.