Mathmatics presentation1
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The document presents the solution to a homogeneous differential equation: dy/dx = y/x + xy It uses the substitution y = vx to transform the equation into a separable equation. The equation is then integrated using partial fractions to obtain the general solution: ln[x + (y/x)] = 2(y/x)^-1/2
![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)](https://image.slidesharecdn.com/mathmatics-presentation1-160731065316/85/Mathmatics-presentation1-5-320.jpg)
Mathmatics presentation1
- 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)