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## Homework Statement

Solve the given differential equation by using an appropriate substitution.

## Homework Equations

[tex]x\frac{dy}{dx} = y + \sqrt{x^{2} - y^{2}}[/tex], x > 0

## The Attempt at a Solution

[tex]x\frac{dy}{dx} = y + \sqrt{x^{2} - y^{2}}[/tex]

[tex]xdy = (y + \sqrt{x^{2} - y^{2}})dx[/tex]

[tex]y = ux[/tex]

[tex]u = \frac{y}{x}[/tex]

[tex]dy = udx + xdu[/tex]

[tex]x[udx + xdu] = (ux + \sqrt{x^{2} - u^{2}x^{2}})dx[/tex]

[tex]xudx + x^{2}du = uxdx + x\sqrt{1 - u^{2}}dx[/tex]

[tex]\frac{du}{\sqrt{1 - u^{2}}} = \frac{dx}{x}[/tex]

[tex]\frac{du}{\sqrt{1 - u^{2}}} - \frac{dx}{x} = 0[/tex]

[tex]\int\frac{du}{\sqrt{1 - u^{2}}} - \int\frac{dx}{x} = 0[/tex]

[tex]sin^{-1}(u) - ln|x| = ln|c|[/tex]

[tex]sin^{-1}(\frac{y}{x}) - ln|x| = ln|c|[/tex]

[tex]e^{sin^{-1}(\frac{y}{x})} - x = c[/tex]

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