2yxe^{x/y} dx + (y-2xe^{x/y} dx = 0

Question 2:

Show that differential equation \displaystyle 2ye^{x/y} dx + (y - 2xe^{x/y}) dy = 0  is homogeneous and find its particular solution, given that x = 0 when y = 1.                             [CBSE(North) 2016; (south) 2016, 2017]

Solution:

Given: \displaystyle 2ye^{x/y} dx + (y - 2xe^{x/y}) dy = 0

\displaystyle \Rightarrow \dfrac{dx}{dy} = -\dfrac{y - 2x e^{x/y}}{2y e^{x/y}}

\displaystyle \Rightarrow \dfrac{dx}{dy} = \dfrac{2x e^{x/y}-y}{2y e^{x/y}}

Let,  \displaystyle F(x , y) = \dfrac{2x e^{x/y}-y}{2y e^{x/y}}

∴     \displaystyle F(\lambda x , \lambda y)= \dfrac{2\lambda x e^{\lambda x/\lambda y}-\lambda y}{2\lambda y e^{\lambda x/\lambda y}}

\displaystyle \Rightarrow F(\lambda x , \lambda y) = \lambda^0\dfrac{2x e^{x/y}-y}{2y e^{x/y}}

\displaystyle \Rightarrow F(\lambda x , \lambda y)  = \lambda^0 F(x , y)

Hence, given differential equation is homogeneous.

Now,

\displaystyle \dfrac{dx}{dy} = \dfrac{2x e^{x/y}-y}{2y e^{x/y}}  .  .  .  .  . . . . (i)

Let,  z = vy

\Rightarrow \dfrac{dx}{dy} =  v + \dfrac{dv}{dy}

From (i)

\displaystyle v + \dfrac{dv}{dy} = \dfrac{2v y e^{v y/y}-y}{2y e^{v y/y}}

\displaystyle \Rightarrow  \dfrac{dv}{dy} = \dfrac{2v e^{v}-1}{2 e^{v}} - v

\displaystyle \Rightarrow  \dfrac{d v}{dy} = \dfrac{2v e^{v}-1 - 2v e^v}{2 e^{v}}`

\displaystyle \Rightarrow  \dfrac{d v}{dy} = \dfrac{-1 }{2 e^{v}}

\displaystyle \Rightarrow 2 e^{v} dv = - dy

\displaystyle \Rightarrow \int 2 e^{v} dv = - \int dy

\displaystyle \Rightarrow 2 e^v = - \log y + C

\displaystyle \Rightarrow 2 e^{x/y} + \log y = C

When x = 0, y = 1

∴   2e^0  + \log 1 = C

\Rightarrow C = 2

Hence the required solution is

\displaystyle  2 e^{x/y} + \log y = 2

Some other question

Question 1:  Solve the following differential equation: (1 + e^{y/x})dy + e^{y/x}(1 - \frac{y}{x}) = 0, x ≠ 0

Solution : For solution click here

Question 3: show that the given differential equation is homogeneous and solve it.

\displaystyle\{x\cos (\frac{y}{x})+y\sin (\frac{y}{x})\}y dx =  \{y\sin (\frac{y}{x}) - x\cos (\frac{y}{x})\}x dy.

Solution: For solution click here

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