Question

1. Determine if the differential equation x^2y′=y(x+y) is homogeneous or Bernouilli or both. Give a solution using any method that applies.

2. Solve the differential equation y′= 2x(y+y^2) using the method of Bernouilli equation. Also give a solution for the same differential equation using the method of separable DE.

3. Consider the differential equation y′′= (y′)^2. It is has both x and y variable missing.Give solutions to the DE

using the two different methods corresponding t ox-variable missing, and y-variable missing.

Answer 1

het en gorge Gap Glazy) - Ilary) A differential equation is homogeneous, if G (x, ay) = Grays. G (ix, ay ) = Ny Carly a yeary _ Gap hence it å homogeneas equation, son dere ce years). Le let 1 - + ta y = tx N2 substituting in equo- trudt = t + t² da dn at - dx

stedt = bn(x)) + c lal) + c - ln (xt) + c 1+C - 2 +1 ln (la)+C t = la Bel) +C We already supposed wczyce ly = center

which is of the form. dy + pers.y = Q(x) y hence it y a besnoulli's equation let ytzt Substituting in egr - da which is linear equation of the form de pour pw.y = QQ. Integrating factor s Polda Sidu I.F= e = e = = 2 enkel e

General solution tx=tlede tx= - Se de t.x= - dendal) + C put y = t in= – le (la) + c اع Ty = capers