Ecuación Diferencial Homogénea

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\begin{gather*} x\frac{dy}{dx} =y+\sqrt{x^{2}} -y^{2} ,x >0\\ \\ \frac{dy}{dx} =\ \frac{y+\sqrt{x^{2}} -y^{2}}{x} \ ;\ x\ >\ 0\\ \\ \frac{dy}{dx} =\ \frac{y+\sqrt{x^{2}} -y^{2}}{x} \ .\ \frac{\frac{1}{x}}{\frac{1}{x}}\\ =\ \frac{\frac{y}{x} +\frac{\sqrt{x^{2-} y^{2}}}{x}}{\frac{x}{x}}\\ =\ \frac{\frac{y}{x} +\frac{\sqrt{x^{2-} y^{2}}}{x^{2}}}{\frac{x}{x}}\\ =\ \frac{\frac{y}{x} +\sqrt{1-\left(\frac{y}{x}\right)^{2}}}{1}\\ \\ \frac{dy}{dx} =\frac{y}{x} +\sqrt{1-\left(\frac{y}{x}\right)^{2}}\\ \\ u\ =\ \frac{y}{x} \Longrightarrow \ y\ =\ ux\ \Longrightarrow \ \frac{dy}{dx} \ =\ u\ +\ x\frac{du}{dx}\\ \\ \frac{dy}{dx} =\ \frac{y}{x} +\sqrt{1-\left(\frac{y}{x}\right)^{2}}\\ \\ u+x\frac{du}{dx} =u+\sqrt{1-( u)^{ \begin{array}{l} 2\\ \end{array}}}\\ \\ x\frac{du}{dx} =u+\sqrt{1-u^{2}} -u\\ \\ x\frac{du}{dx} =\sqrt{1-u^{2}}\\ \\ x\frac{du}{dx} =\sqrt{1-u^{2}}\\ \Longrightarrow \ \frac{du}{\sqrt{1-u^{2}}} =\ \frac{dx}{x}\\ \end{gather*}

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