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Question
Show that the given differential equation is homogeneous and solve them.
`x dy - y dx = sqrt(x^2 + y^2) dx`
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Solution
`x dy - y dx = sqrt(x^2 + y^2) dx`,
Which can be written as `x dy/dx = y + sqrt (x^2 + y^2)`
or `dy/dx = y/x + sqrt (1 + (y/x)^2)` ....(1)
Since R.H.S. is of the form `g(y/x)`, and so 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`
Substituting the value of y and `dy/dx` in (1), we get
`v + x (dv)/dx = v + sqrt (1 + v^2)`
⇒ `x (dv)/dx = sqrt(1 + v^2)`
⇒ `dx/x = (dv)/sqrt(1 + v^2)`
⇒ `int dx/x = int (dv)/ sqrt(1 + v^2)`
⇒ `log x + log C_1 = log |v + sqrt (1+ v^2)|`
⇒ `log x + log C_1 = log |y/x + sqrt (1 + y^2/x^2)|`
⇒ `log C_1 x = log |y + sqrt (x^2 + y^2)| - log x`
⇒ `pm C_1 x^2 = y + sqrt (x^2 + y^2)`
⇒ `Cx^2 = y + sqrt (x^2 + y^2)`
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