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

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

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a^2dx=xsqrt(x^2-a^2)dy

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

y = a arctan (\frac{x ^{2} - a ^{2}}{a}) + c_{1}

Solution steps

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

Let

y

be the dependent variable. Divide by

d x

a^{2} = x x^{2} - a^{2} \frac{d y}{d x}

Substitute

\frac{d y}{d x}

with

y^{'} (x)

a^{2} = x x^{2} - a^{2} y^{^{'}} (x)

Isolate

y^{'} (x) : y^{'} (x) = \frac{a ^{2}}{x x ^{2} - a ^{2}}

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

then

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

y = \int \frac{a ^{2}}{x x ^{2} - a ^{2}} d x

\int \frac{a ^{2}}{x x ^{2} - a ^{2}} d x = a arctan (\frac{x ^{2} - a ^{2}}{a}) + c_{1}

y = a arctan (\frac{x ^{2} - a ^{2}}{a}) + c_{1}

e^{2 t} sin (3 t)

y = 3 x, y = 9, y = \frac{1}{4} x^{2}

\int (x^{3} + 2)^{8} 3 x^{2} d x

\frac{d}{d x} (5 x - x)

x \to - \frac{π}{2} lim (csc (x))

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