What is the general solution for solvefor y,ln(x)=arctan(y/x) ?

The general solution for solvefor y,ln(x)=arctan(y/x) is y=tan(ln(x))x

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solvefor y,ln(x)=arctan(y/x)

solve for

y, ln (x) = arctan (\frac{y}{x})

y = tan (ln (x)) x

Solution steps

ln (x) = arctan (\frac{y}{x})

Subtract

arctan (\frac{y}{x})

from both sides

ln (x) - arctan (\frac{y}{x}) = 0

Let:

u = \frac{y}{x}

ln (x) - arctan (u) = 0

Move

ln (x)

to the right side

ln (x) - arctan (u) = 0

Subtract

ln (x)

from both sides

ln (x) - arctan (u) - ln (x) = 0 - ln (x)

Simplify

- arctan (u) = - ln (x)

- arctan (u) = - ln (x)

Divide both sides by

- 1

- arctan (u) = - ln (x)

Divide both sides by

- 1

\frac{- arctan ( u )}{- 1} = \frac{- ln ( x )}{- 1}

Simplify

arctan (u) = ln (x)

arctan (u) = ln (x)

Apply trig inverse properties

arctan (u) = ln (x)

arctan (x) = a \Rightarrow x = tan (a)

u = tan (ln (x))

u = tan (ln (x))

Substitute back

u = \frac{y}{x}

\frac{y}{x} = tan (ln (x)) : y = tan (ln (x)) x; x \neq = 0

\frac{y}{x} = tan (ln (x))

Multiply both sides by

x

\frac{y}{x} = tan (ln (x))

Multiply both sides by

x

\frac{y x}{x} = tan (ln (x)) x; x \neq = 0

Simplify

y = tan (ln (x)) x; x \neq = 0

y = tan (ln (x)) x; x \neq = 0

y = tan (ln (x)) x

cos (a) + 1 = 4 cos (a) + 1

6 cos (x) + 3 sin (x) = 5

(2 sin (x) - 1) cos (x) = 0

sin (2 x) + cos (x) = 0, x \leq 2 π, 0

6 sin (x) = 5 cos (x)

What is the general solution for solvefor y,ln(x)=arctan(y/x) ?
The general solution for solvefor y,ln(x)=arctan(y/x) is y=tan(ln(x))x