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Maiclubk

Answered question

2020-11-24

Give the correct answer and solve the given equation

(x - y) d x + (3 x + y) d y = 0, when x = 3, y = - 2

Sally Cresswell

Skilled2020-11-25Added 91 answers

The given diferential equation is

(x - y) d x + (3 x + y) d y = 0,

with

x = 3, y = 2

Now

(x - y) d x + (3 x + y) d y = 0, \Rightarrow \frac{d y}{d x} = \frac{y - x}{3 x = y}

since the given differential equation is a homogenuous equation, putting

y = v x

, we have

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

So the equation becomes.

v + x d \frac{v}{d x} = \frac{v - 1}{v + 3}

\Rightarrow x d \frac{v}{d x} = \frac{v - 1}{v + 3} - v = - \frac{{(v + 1)}^{2}}{v + 3}

\Rightarrow \frac{v + 3}{{(v + 1)}^{2}} d v + \frac{d x}{x} = 0

\Rightarrow \frac{v + 1 + 2}{{(v + 1)}^{2}} d v + \frac{d x}{x} = 0

\Rightarrow d \frac{v}{(v + 1)} + \frac{2 d v}{{(v + 1)}^{2}} + \frac{d x}{x} = 0

\Rightarrow \int \frac{d x}{(v + 1)} + \int \frac{2 d v}{{(v + 1)}^{2}} + \int \frac{d x}{x} = C

where C is a integrating constant

\Rightarrow \ln (v + 1) = \frac{2}{(v + 1)} + \ln (x) = C

\Rightarrow \ln (x (v + 1)) - \frac{2}{(v + 1)} = C

putting

v = \frac{y}{x}

we have

\Rightarrow \ln (y + x) - 2 \frac{x}{(x + y)} = C

Since

x + 3, y + 2

, we have

C = \ln 5 - \frac{6}{5}

Hence the solution of the equation is

\Rightarrow \ln (y + x) - 2 \frac{x}{(x + y)} = \ln 5 - \frac{6}{5}

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