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  4. The number of solutions of (dy/dx)=(y+1/x-1) when y(1)=2 is

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Q. The number of solutions of $\frac{dy}{dx}=\frac{y+1}{x-1}$ when $y\left(1\right)=2$ is

Differential Equations

Solution:

$\frac{dy}{dx}=\frac{y+1}{x-1}$
$\Rightarrow \frac{dy}{y+1}=\frac{dx}{x-1}$
On integration, we get
$log\left(y + 1\right) + logc = log\left(x - 1\right)$
$\Rightarrow \left(y + 1\right)c = \left(x - 1\right)$
Now, $y\left(1\right) = 2$
$\Rightarrow 3c=0$
$\Rightarrow c=0$
$\therefore x - 1 = 0$
$\Rightarrow x=1$
Hence, only one solution exists.