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WELCOME TO MY
PRESENTATION
MY PRESENTATION TOPIC IS
HOMOGENEOUS
NAME: PIAS AHMMED
ID:151-15-5198
SOLVE THE FOLLOWING HOMOGENEOUS DE
𝑑𝑦
𝑑𝑥
=
𝑦
𝑥+ 𝑥𝑦
Sol:
The given DE is
𝑑𝑦
𝑑𝑥
=
𝑦
𝑥+ 𝑥𝑦
Let, f(x,y)=
𝑦
𝑥+ 𝑥𝑦
f(tx,ty)=𝑡0(
𝑦
𝑥+ 𝑥𝑦
)
=𝑡0 𝑓(𝑥, 𝑦)
f(x , y) is a homogenous function of Zero degree. So the given DE is also a homogenous DE.
Let us consider y= vx
Or,
𝑑𝑦
𝑑𝑥
= v+x
𝑑𝑦
𝑑𝑥
Or,
𝑣𝑥
𝑥+ 𝑥2 𝑣
= v+x
𝑑𝑣
𝑑𝑥
Or,
𝑣𝑥
𝑥(1+ 𝑣)
=v+ x
𝑑𝑣
𝑑𝑥
Or,
𝑣
1+ 𝑣
= 𝑣 + 𝑥
𝑑𝑣
𝑑𝑥
Or, 𝑣 + 𝑥
𝑑𝑣
𝑑𝑥
=
𝑣
1+ 𝑣
Or,𝑥
𝑑𝑣
𝑑𝑥
=
𝑣
1+ 𝑣
− 𝑣
Or, 𝑥
𝑑𝑣
𝑑𝑥
=
𝑣−𝑣−𝑣. 𝑣
1+ 𝑣
Or, 𝑥
𝑑𝑣
𝑑𝑥
=
−𝑣 𝑣
1+ 𝑣
Separating the variables and integrating
Or, (
1+ 𝑣
−𝑣. 𝑣
)𝑑𝑣=
1
𝑥
𝑑𝑥
Or,(
1
−𝑣 𝑣
−
𝑣
𝑣 𝑣
)𝑑𝑣=
1
𝑥
𝑑𝑥
Or,
1
−𝑣 𝑣
𝑑𝑣 −
1
𝑣
𝑑𝑣=
1
𝑥
𝑑𝑥
Or, 1𝑑𝑣 −
1
𝑣
𝑑𝑣=
1
𝑥
𝑑𝑥
Or, -v-lnv=lnx
Or, -v=lnvx
Or,-
𝑦
𝑥
=lny
THANK YOU
ALL….!!!

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Higher order math

  • 2. MY PRESENTATION TOPIC IS HOMOGENEOUS NAME: PIAS AHMMED ID:151-15-5198
  • 3. SOLVE THE FOLLOWING HOMOGENEOUS DE 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥+ 𝑥𝑦 Sol: The given DE is 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥+ 𝑥𝑦 Let, f(x,y)= 𝑦 𝑥+ 𝑥𝑦 f(tx,ty)=𝑡0( 𝑦 𝑥+ 𝑥𝑦 ) =𝑡0 𝑓(𝑥, 𝑦) f(x , y) is a homogenous function of Zero degree. So the given DE is also a homogenous DE.
  • 4. Let us consider y= vx Or, 𝑑𝑦 𝑑𝑥 = v+x 𝑑𝑦 𝑑𝑥 Or, 𝑣𝑥 𝑥+ 𝑥2 𝑣 = v+x 𝑑𝑣 𝑑𝑥 Or, 𝑣𝑥 𝑥(1+ 𝑣) =v+ x 𝑑𝑣 𝑑𝑥 Or, 𝑣 1+ 𝑣 = 𝑣 + 𝑥 𝑑𝑣 𝑑𝑥 Or, 𝑣 + 𝑥 𝑑𝑣 𝑑𝑥 = 𝑣 1+ 𝑣 Or,𝑥 𝑑𝑣 𝑑𝑥 = 𝑣 1+ 𝑣 − 𝑣 Or, 𝑥 𝑑𝑣 𝑑𝑥 = 𝑣−𝑣−𝑣. 𝑣 1+ 𝑣 Or, 𝑥 𝑑𝑣 𝑑𝑥 = −𝑣 𝑣 1+ 𝑣
  • 5. Separating the variables and integrating Or, ( 1+ 𝑣 −𝑣. 𝑣 )𝑑𝑣= 1 𝑥 𝑑𝑥 Or,( 1 −𝑣 𝑣 − 𝑣 𝑣 𝑣 )𝑑𝑣= 1 𝑥 𝑑𝑥 Or, 1 −𝑣 𝑣 𝑑𝑣 − 1 𝑣 𝑑𝑣= 1 𝑥 𝑑𝑥 Or, 1𝑑𝑣 − 1 𝑣 𝑑𝑣= 1 𝑥 𝑑𝑥 Or, -v-lnv=lnx Or, -v=lnvx Or,- 𝑦 𝑥 =lny