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  4. The solution of the differential equation (dy/dx)=(x2+y2+1/2xy) satisfying y( 1) = 1, is

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Q. The solution of the differential equation $\frac{dy}{dx}=\frac{x^{2}+y^{2}+1}{2xy}$ satisfying $y( 1) = 1$, is

Differential Equations

Solution:

Given, $\frac{dy}{dx}=\frac{x^{2}+y^{2}+1}{2xy}$
$\Rightarrow 2xydy=\left(x^{2}+y^{2}+1\right)dx$
$\Rightarrow 2xydy-y^{2}dx=\left(x^{2}+1\right)dx$
$\Rightarrow xd\left(y^{2}\right)-y^{2}dx=\left(x^{2}+1\right)dx$
$\Rightarrow \frac{xd\left(y^{2}\right)-y^{2}\,dx}{x^{2}}=\left(1+\frac{1}{x^{2}}\right)dx$
$\Rightarrow d\left(\frac{y^{2}}{x}\right)=d\left(x-\frac{1}{x}\right)$
$\Rightarrow \frac{y^{2}}{x}=x-\frac{1}{x}+C$
$\Rightarrow y^{2}=x^{2}-1+Cx$
$\Rightarrow y^{2}=\left(x+\frac{C}{2}\right)^{2}-1-\frac{C^{2}}{4}$
Clearly, it represents a hyperbola.