Show that the given differential equation is homogeneous and solve them. y dx+xlog(yx)dy-2x dy=0 - Mathematics

Question

Show that the given differential equation is homogeneous and solve them.

`y dx + x log(y/x)dy - 2x dy = 0`

Sum

Solution

`y dx + x log (y/x) dy - 2x dy = 0`

⇒ `y/x + log (y/x) dy/dx - 2 dy/dx = 0`

⇒ `dy/dx = (y/x)/(2 - log (y/x))` .....(1)

Since R.H.S. is of the form `g(y/x)`, it is a homogeneous function of degree zero.

Therefore equation (1) is a homogeneous differential equation.

To solve this, put y = vx

⇒ `dy/dx = v + x (dv)/dx,` then (1) becomes

`v + x (dv)/dx = v/ (2 - log v)`

⇒ `x (dv)/dx = v/ (2 - log v) - v`

⇒ `((2 - log v)dv)/(v log v - v) = dx/x`

⇒ `(1 - (log v - 1))/(v (log v - 1)) dv = dx/x` .....(2)

Integrating (2) both sides, we get

`int (1/ (v (logv - 1)) - 1/v) dv`

`= int dx/x + C_1`

⇒ `int (1/v)/(log v - 1) dv - int 1/v dv = int dx /x + C_1`

⇒ `log |log v - 1| - log |v|`

= log |x| + C₁

⇒ `log |(log v - 1)/(v x)| = C_1`

⇒ `|(log v - 1)/ (v x)| = e^(C_(1))`

⇒ `(log v - 1)/(v x) = pm e^(C_(1)) = C` (say)

⇒ `log (y/x) - 1 = Cy` ...`(∵ v = y/x)`

which is the required general solution.

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Methods of Solving First Order, First Degree Differential Equations - Homogeneous Differential Equations

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Chapter 9: Differential Equations - Exercise 9.5 [Page 406]

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NCERT Mathematics Part 1 and 2 [English] Class 12

Chapter 9 Differential Equations
Exercise 9.5 | Q 9 | Page 406

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