Question

Solve the following D.E. (3.+ 2y)dx + (4.xy + 6y2)dy = 0 Solve the following D.E. (x²y)dx + y(x3 +e-3y sin y)dy = 0

Can you solve these dif. equations?

Answer 1

* (32? + 2y?) do 4 (4x4 46 4? Jdy =0. compare with, M dx + N dy = 0 IM= 3x? tay? am Of 44 = 44 SM = 4x4 + 6y? ON - 44 +6 =44 here m an so, D.E. is exact 776 general solution is, I m dict Tyr constant) , No dy :( (without a term? Je3x3 + 2 }) dx + , Gy? dy =( 12 , 24 25 26 ]

☆ (a?y) dx + y (x3 + e 34. siny) dy =0 (23) dx + ca 3y + ye 34 siny) dy 50 compare with, M dx t Ndy to -34 M= acly N= ?y tye 39 siny 2N = 384 to 7x here and I am so, de is not exact So, find integration factor If : Stendy , where fal = f(= saya?

113-4) dy IFE 34- eny e- Festeniny 34 Inn IF a multiply both sides. Fre) - mulhey both sides. (xi) dz + 4.4(20 + e.siny) 2y = (23 (34) do + 33 34 + siny) dy to compane with, M dx t N dy = 0,

N = d?e34 t siny 2 34 Μ- ιο 2,34.3 JM ge. d = 3 oc'e39 to 3 34 7M IN 2Y 20 Now EIN » so, DE is exact, A gx general solution is = C | M dx + [ N dy. (Yr constant) (without a-term) I 23 34 dx + J siny dy ac €34.2? + (-6054) = 0 34 -Cosy ECR answer