#31 Measure Your Calibre

Relevant wiki: Bernoulli Equation

$\begin{aligned} \frac {dy}{dx} & = \frac {x^2+y^2+1}{2xy} \\ \frac {dy}{dx} - \frac y{2x} & = \frac {x^2+1}{2xy} & \small \color{#3D99F6} \text{Bernoulli equation: }y' + p(x)y = q(x)y^n \\ \frac {d}{dx}\sqrt v - \frac {\sqrt v}{2x} & = \frac {x^2+1}{2x\sqrt v} & \small \color{#3D99F6} \text {Let } v=y^{1-n} = y^2, \ y = \sqrt v \\ \frac 1{2\sqrt v} \cdot \frac {dv}{dx} - \frac {\sqrt v}{2x} & = \frac {x^2+1}{2x\sqrt v} & \small \color{#3D99F6} \text{Multiplying both sides by }2\sqrt v \\ \frac {dv}{dx} - \frac vx & = \frac {x^2+1}x & \small \color{#3D99F6} \text{First-order ODE: } v' + vP(x) = Q(x) \end{aligned}$

$\begin{aligned} \implies v & = \frac {\int \exp \left(- \int \frac {dx}x \right) \cdot \frac {x^2+1}x \ dx+C}{\exp \left(-\int \frac {dx}x \right)} & \small \color{#3D99F6} v = \frac {\int e^{\int P(x) \ dx} Q(x) \ dx+C}{e^{\int P(x) \ dx}} \\ y^2 & = \frac {\int \frac {x^2+1}{x^2} dx + C }{\frac 1x} & \small \color{#3D99F6} \text{where }C \text{ is the constant of integration.} \\ & = x^2 - 1 + Cx & \small \color{#3D99F6} \text{Putting }y(1) = 1 \\ 1 & = 1-1+C \\ \implies C & = 1 \\ \implies y^2 & = x^2 - 1 + x \\ & = \boxed{x(1+x)-1} \end{aligned}$