Question

T2-1 (20 Points): Find the P.S. of the IVP: x2 + 2xy + y2 1 + (x + y)2 y(x = 0) = 4 Primes denote derivatives WRT x. Sy' T2-2 (20 Points): Find the G.S. of the DE: xy' + y = 3x2 Prime denotes derivative WRT x. (Hint: guess a P.S. yı = Axa)

Answer 1

Q. Answer:-) Given y = (x² + 2xy + y² yo=4 1+0+ y = x+4) 1+ (x+y), dt dx t Ht² x+y=t differentiating with Adt – – + + Itt -*++* respect to 'x' on both sides. we get It dy =dt an dt कर 1 + x² -(dt dn put this in BU+tJH = dne Integrating 40)

at² dt = (dx + c ${+++) = x+ where cry an integral constant : 1x dx = xh7 Hu ntl t ६ ego becomes (x+y) + (x+y) = x + c whing y(x=0) = 4 (+ YOU) = (0+ YO) = 0 + 4 x 4 = 6 1+=cata? Hence perticular solution is 4. + 3

=342 Q. 2. Answer:-) Given. ryl ty how let's write it in the Standard form du -py P4 Q are functions of (le dy dyt Where y + Ty = 3x² x =) dy tl y = 3x dn a ( Which is a first order linean differential equation.) combring it with standard formi we have P = I &Q=3x. (pon Integrating factor I-F.) = e how 7

Lide TO Inn = e бо I it. - =n dy +Pouy Recall: General solution of + Powy = k) is given by y XI •fo = f I . F • X. Qu de to da integral constant. So y x x = ( x x x dx fe Home Q&u=3% 2 3 Yx = 3x² da te y x = 3,43 + c 4 x = = x + c x + c Ly Answer b OR x + cxt