Question

El solve - 2x² dy = y(y² + 32²) dx . sol: let us go e 131 Find the general sal of ODE. xy" - (x + 1) yl ty=o given that y, is a sal. sal y=ay, +CY₂ . cé + c2(x+1). W Freud the general sal for: y y Cyry'). soli 3-missing. u-ylinde yo 51 solve: y" – Gy! +9y -- .

Please can you help me to solve these problems

Answer 1

288dy = y 192 +3 x²) de a = y + 3x4 o 223 let y = ux = ut xdu Put this in eq (1) utx dy = 43113 + 3 - dx 223 x² (4x) dx - 2 4 + x dy = 43+ 34 xdy = 43 +34-24 dx 2 de ou + de de = L (U+W)

tot let -2 zate = I v du = die 42 43 If a lle dx = elogx = x solution is u. x = (xx-1 die + A u.x = x+ K 2 y = (-X+*) = - + c = -x + x

xy" - (x + 1) y't y = 0 By comparing above equation with P(x) y" + q (oly' + r(x)y=o, we get Q(x) = -(x+1) and here f(x)= ex second solution - ver where va ser p exp[-{{q(X) / P(x)} dx dx -0. f(0)}2 Now he dx = - 1 st 1 dx = (1 + 2 ) die = (x+ logx) exp[-]{2(x)/P(x)} dre7 = exp(x+10g1e) = xt logue - ex lo gexer from (1) 20 e Er -{{1x lengde Required second solution_vex = -xer = -xexe / _- (311)

So ral son is Y=c, Yi+ 4% = cientą (x+1) - [-sign will go in a

y" = y(y+y) - da putting in x du = XL (lety) - ③ ut y =v so from ear ② dv - V du = dy vti log (U+1) = y +x (integrating) (log (uty + 1) = y + x| (putting us to dy + y + 1-ce

dy = x + C2 Icey y not solution After this, integration is manually possible, so leave till equation 3 y" - 64l +94 = 0 auxi. egy- m-6m +9=0 m = 3,3 So soll is 32 y = (cit (x) en
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