Solve the following differential equation: dy dx = (x-y)+3 2(x-y)… - Vidyadip

Question

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{(\text{x}-\text{y})+3}{2(\text{x}-\text{y})+5}$

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Answer

We have,
$\frac{\text{dy}}{\text{dx}} = \frac{(\text{x}-\text{y} +3)}{2(\text{x}-\text{y})+5}$
Putting $\text{ x} - \text{y} = \text{v}$
$\Rightarrow 1 - \frac{\text{dy}}{\text{dx}} = \frac{\text{dv}}{\text{dx}}$
$\Rightarrow \frac{\text{dy}}{\text{dx}} = 1 - \frac{\text{dv}}{\text{dx}}$
$\therefore 1 - \frac{\text{dv}}{\text{dx}} = \frac{\text{v} + 3}{2\text{v} + 5}$
$\Rightarrow \frac{\text{dv}}{\text{dx}} = 1 - \frac{\text{v} + 3}{2\text{v} + 5}$
$\Rightarrow \frac{\text{dv}}{\text{dx}} = \frac{2\text{v} + 5 - \text{v} - 3}{2\text{v} + 5}$
$\Rightarrow \frac{\text{dv}}{\text{dx}} = \frac{\text{v}+2}{2\text{v}+5}$
$\Rightarrow \frac{2\text{v}+5}{\text{v}+2}\text{dv} = \text{dx}$
Integrating both sides, we get
$\int \frac{2\text{v}+5}{\text{v}+2}\text{dv} = \int\text{dx}$
$\Rightarrow \int \frac{2\text{v}+4+1}{\text{v}+2}\text{dv} = \int \text{dx}$
$\Rightarrow \int\Big(\frac{2\text{v}+4}{\text{v}+2} + \frac{1}{\text{v}+2}\Big)\text{dv} = \int\text{dx}$
$\Rightarrow 2\int\text{dv}+\int\frac{1}{\text{v}+2}\text{dv} = \int \text{dx}$
$\Rightarrow2\text{v}+\log |\text{v}+2| = \text{x}+\text{C}$
$\Rightarrow 2 (\text{x}-\text{y})+\log|\text{x}-\text{y}+2| = \text{x}+\text{C}$

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