What is the solution for dx=xsqrt(x^2-16dy) ?

The solution for dx=xsqrt(x^2-16dy) is y=-1/16 (-1/x-(x^3)/3)+c_{1}

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dx=xsqrt(x^2-16dy)

d x = x x^{2} - 16 d y

y = - \frac{1}{16} (- \frac{1}{x} - \frac{x ^{3}}{3}) + c_{1}

Solution steps

d x = x x^{2} - 16 d y

Let

y

be the dependent variable. Divide by

d x

1 = x x^{2} - 16 \cdot \frac{d y}{d x}

Substitute

\frac{d y}{d x}

with

y^{'} (x)

1 = x x^{2} - 16 y^{^{'}} (x)

Isolate

y^{'} (x) : y^{'} (x) = - \frac{1 - x ^{4}}{16 x ^{2}}

f^{^{'}} (x) = g (x)

then

f (x) = \int g (x) d x

y = \int - \frac{1 - x ^{4}}{16 x ^{2}} d x

\int - \frac{1 - x ^{4}}{16 x ^{2}} d x = - \frac{1}{16} (- \frac{1}{x} - \frac{x ^{3}}{3}) + c_{1}

y = - \frac{1}{16} (- \frac{1}{x} - \frac{x ^{3}}{3}) + c_{1}

What is the solution for dx=xsqrt(x^2-16dy) ?
The solution for dx=xsqrt(x^2-16dy) is y=-1/16 (-1/x-(x^3)/3)+c_{1}