What is the general solution for solvefor y,z=x^2sin(xy^2) ?

The general solution for solvefor y,z=x^2sin(xy^2) is

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solvefor y,z=x^2sin(xy^2)

Solution

solve for

y, z = x^{2} sin (x y^{2})

Solution

y = \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}, y = \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

Solution steps

z = x^{2} sin (x y^{2})

Switch sides

x^{2} sin (x y^{2}) = z

Divide both sides by

x^{2}; x \neq = 0

x^{2} sin (x y^{2}) = z

Divide both sides by

x^{2}; x \neq = 0

\frac{x ^{2} sin ( x y ^{2} )}{x ^{2}} = \frac{z}{x ^{2}}; x \neq = 0

Simplify

sin (x y^{2}) = \frac{z}{x ^{2}}; x \neq = 0

sin (x y^{2}) = \frac{z}{x ^{2}}; x \neq = 0

Apply trig inverse properties

sin (x y^{2}) = \frac{z}{x ^{2}}

General solutions for

sin (x y^{2}) = \frac{z}{x ^{2}}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x y^{2} = arcsin (\frac{z}{x ^{2}}) + 2 πn, x y^{2} = π + arcsin (- \frac{z}{x ^{2}}) + 2 πn

x y^{2} = arcsin (\frac{z}{x ^{2}}) + 2 πn, x y^{2} = π + arcsin (- \frac{z}{x ^{2}}) + 2 πn

Solve

x y^{2} = arcsin (\frac{z}{x ^{2}}) + 2 πn : y = \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}; x \neq = 0

x y^{2} = arcsin (\frac{z}{x ^{2}}) + 2 πn

Divide both sides by

x; x \neq = 0

x y^{2} = arcsin (\frac{z}{x ^{2}}) + 2 πn

Divide both sides by

x; x \neq = 0

\frac{x y ^{2}}{x} = \frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

Simplify

y^{2} = \frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

y^{2} = \frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

y = \frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}, y = - \frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

Simplify

\frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

\frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}

Combine the fractions

\frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

= \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

Simplify

- \frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : - \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

- \frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}

Combine the fractions

\frac{arcsin ( \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

= - \frac{2 πn + arcsin ( \frac{z}{x ^{2}} )}{x}

= - \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}

y = \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}; x \neq = 0

Solve

x y^{2} = π + arcsin (- \frac{z}{x ^{2}}) + 2 πn : y = \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}; x \neq = 0

x y^{2} = π + arcsin (- \frac{z}{x ^{2}}) + 2 πn

Divide both sides by

x; x \neq = 0

x y^{2} = π + arcsin (- \frac{z}{x ^{2}}) + 2 πn

Divide both sides by

x; x \neq = 0

\frac{x y ^{2}}{x} = \frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

Simplify

y^{2} = \frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

y^{2} = \frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

y = \frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}, y = - \frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}; x \neq = 0

Simplify

\frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

\frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}

Combine the fractions

\frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

= \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

Simplify

- \frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : - \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

- \frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x}

Combine the fractions

\frac{π}{x} + \frac{arcsin ( - \frac{z}{x ^{2}} )}{x} + \frac{2 πn}{x} : \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

= - \frac{π + 2 πn + arcsin ( - \frac{z}{x ^{2}} )}{x}

= - \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

y = \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}; x \neq = 0

y = \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{arcsin ( \frac{z}{x ^{2}} ) + 2 πn}{x}, y = \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}, y = - \frac{π + arcsin ( - \frac{z}{x ^{2}} ) + 2 πn}{x}

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor y,z=x^2sin(xy^2) ?
The general solution for solvefor y,z=x^2sin(xy^2) is