Question

I need help with question's 1 and 2

T2-1 (20 Points): Find the P.S. of the IVP: x2 + 2xy + y2 y = 1+ (x + y)2 y(x = 0) = 4 Primes denote derivatives WRT X. 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 = Ax")

Answer 1

dz => 1 + dy dn da dz dz du da Now from (ii) we have dirodz Z? -1 - dr 1 +2² dz dn 22 1+22 dz 1+2²-22 da 1 +22 dz - da 1 +22 -> (1+22) dz = dx Taking Integration (iii) we in both sides of have 10+2%dz = forme » / 22 / 2d2= (dx

> 2+ 3 = x+c =rt C 3 (x+y)} => (x+y)+ where Cl being arbitrary constant Now Using Initial condition y (n =o) = 4 that is we Put x=0 and y=4 in the equation we have (0 + 4) + Song) - c. to (iv) 3 4 t 64 3 = C, 12+64 Ci 3 Ci ㅋ 76 3 Now from (n+ y) + -(iv) we have (+3 76 3 3 = at 3

Now, Sede IF = e Jan MI e 11 e login/ е Inl = EU d Multiplying both sides of the equation -(11) by x We have (ay) = 3x² Integrating we have 3 AC da xy 3 ta 2 11 - ry x2 t Ci where C, being arbitrary constant.

at Ci be the General solution of the differential equation ny' +y = 3x2 is the Also given Y, = Axa Particular solution of Then y, must satisfies the differential equation Now, ny, +y = 3x² (iii) Now J = Ana yi & An Q-1 Now (iii) from (&A 20-1) + Ax* = 3x² a = a Ana & And = 3x² - Ax (x + 1) = 3x²

This equality satisfies when x=2 and A- : y = x2 is the required particular solution of the differential equation xy' + y = 3x2 Ans