math

1. HOMOGENEOUS DE
Presented by :
MD.Hasnat Shoheb
ID : 152-15-5813
Daffodil International University
Department of CSE

2. HOMOGENEOUS DE
Question :
𝒅𝒚
𝒅𝒙
=
𝒚
𝒙+ 𝒙𝒚
SOLUTION : let us consider, y = vx ,
𝑑𝑦
𝑑𝑥
= 𝑣 + 𝑥
𝑑𝑣
𝑑𝑥
𝑣𝑥
𝑥+ 𝑥𝑣𝑥
= v + x
𝑑𝑣
𝑑𝑥
𝑣𝑥
𝑥+ 𝑥2 𝑣
= v+𝑥
𝑑𝑣
𝑑𝑥

3. SOLUTION :
𝑣𝑥
𝑥(1+ 𝑣)
= v + 𝑥
𝑑𝑣
𝑑𝑥
𝑣
1+√𝑣
− 𝑣 = 𝑥
𝑑𝑣
𝑑𝑥
−𝑣.𝑣
1
2
1+ √𝑣
= x
𝑑𝑣
𝑑𝑥
1
𝑥
𝑑𝑥 = −
1+ 𝑣 𝑑𝑣
𝑣
3
2

4. SOLUTION :
lnx = -
1
𝑣
3
2
dv -
𝑣
1
2
𝑣
3
2
dv
lnx = - 𝑣
−3
2 dv - 𝑣
1−3
2 dv
lnx = -
𝑣
−3+2
2
−3+2
2
- ln v
lnx = -
𝑣
−1
2
−1
2
- ln v
lnx = 2𝑣
−1
2 - ln v

5. SOLUTION :
ln x + ln v = 2𝑣
−1
2
ln (x+v) = 2 (
𝑦
𝑥
)
−1
2
ln [x+
𝑦
𝑥
] =2 (
𝑦
𝑥
)
−1
2
(Ans)