Question

1. Solve by making a substitution to reduce the given second order De to a first order DE. 1. x? y" + 2xy' - 1 = 0, x>0 (ans. y = Cix-1 + 2 + In x) 2. y" + y(y')} = 0 (ans. 1/3 y3 - 2c1y + C2 = 2x) 3. y'y” = 2, y(0) = 1, y'(0) = 2 (ans. y = 4/3 (x + 1)3/2 - 1/3)

Answer 1

bolly! + 2xy'-1=0, x70 y'=v ixy=v det, x -Utsul 12 - y = 0 16)+ 4.0 . Depy" = -4+x60' putting values from & & 6 in givens egra, GU') +2(V)-10 5) Ocu's -26++ du -vt doc 20 do -1+1 dal su integrating both side, =) - log (-u+1) = loga + loga (4: constant) 17, -vt e 1 - IC, -(y' +1 = -x²64'+aC,= 1 dy doc 1-xG -xa ㅗ GXY X integrating both side,

say = (t dx- & Site de - y loga ( + ((s: constant) => y = logx + 43'+ y= 4x + co + lux Q y"+ y (y)² = 0 let y'=4 ulou y"- du du dy = dol dy da i equation is, uu't yu=0 let uto (since uso is absurd cond) u't yu²=0 ul =) =-y =) . du - y dy integrating both side, (a: constant) - y² + 4 u 4 42 - 26 -24 o ano 224 = (y-24) dy= 2da integrating bath side,

312 g Sy-26) dy = $20x y3 - 244 + 200 3 3 y'y"=2 y(0)=1 Y(0) = 2 det y'=2 - y"=2 - 22=2 2d2= 2dx untegrating both side, -> = 2x+G 2= 54x+84 = y'= 54x+ga (: y'=2) 'yo = 9 iac S 40+99 =) Go :: dy = 54x+ 4 da integrating both side, - Say=& = $ f sacri dae (241) y = + * YO=1 (= 1-2-3 = 1 ... WIL 3 - :) y = $(x +11982