Solve the differential equations condition $$\dfrac{dy}{dx} = \dfrac{(x-y)+3}{2(x-y)+5}$$

Asked by Sakshi | 1 year ago |  53

##### Solution :-

Let us assume that, x – y = v

Differentiating with respect to x on both side, we get,

$$1 – (\dfrac{dy}{dx}) = \dfrac{dv}{dx}$$

Transposing,

$$\dfrac{dy}{dx} = 1 – (\dfrac{dv}{dx})$$ … [equation (ii)]

Substituting equation (ii) in equation (i),

Then,

$$1 – (\dfrac{dv}{dx}) = \dfrac{(v + 3)}{(2v + 3)}$$

Transposing we get,

$$\dfrac{dv}{dx} = \dfrac{1 – (v + 3)}{(2v + 5)}$$

$$\dfrac{dv}{dx} = \dfrac{(2v + 5 – v – 3)}{(2v + 5)}$$

$$\dfrac{dv}{dx} =\dfrac{ (v + 2)}{(2v + 5)}$$

Now, taking like variables on same side,

$$\dfrac{2v + 5}{(v + 2)}dv = dx$$

$$\dfrac{(2v + 4) + 1)}{(v + 2)} dv = dx$$

On dividing we get,

$$2 + \dfrac{1}{(v + 2)}dv = dx$$

Integrating on both side we get,

$$∫2 + \dfrac{1}{(v + 2)} dv = ∫dx$$

We know that,

$$∫\dfrac{dx}{x} = log x + c \;and\; ∫adx = ax + c$$

2v + log [v + 2] = x + c

2(x – y) + log [x – y – 2] = x + c

Answered by Aaryan | 1 year ago

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