"(x−y)dx+xdy=0 Please, solve by using an appropriate substitution." was answered by our community

Emeli Hagan

Answered question

2021-09-04

(x - y) d x + x d y = 0

Please, solve by using an appropriate substitution.

falhiblesw

Skilled2021-09-05Added 97 answers

Given differential equation can be written as:

\frac{d y}{d x} = \frac{y - x}{x} = \frac{y}{x} - 1

Let,

y = υ x

. Then

\frac{d y}{d x} = υ + x d \frac{υ}{d x}

Substituting

y = υ x

in the given equation we have:

υ + x d \frac{υ}{d x} = υ - 1

implies

x d \frac{υ}{d x} = - 1

implies

d υ = - \frac{d x}{x}

implies

\int d υ = - \int \frac{d x}{x}

(we take integration both side)
implies

υ = - \ln x + c

, (c is an integrating constant)
implies

\frac{y}{x} = - \ln x + c

, (we put

υ = \frac{y}{x}

)
implies

y = x (c - \ln x)

The solution of

(x - y) d x + x d y = 0 i s y = x (c - \ln x)

, where c is an integrating constant

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