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Fraction cross multiplication Calculator

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1

Solved example of fraction cross multiplication

$\frac{dy}{dx}=-\frac{x}{y}$
2

Multiplying the fraction and term

$\frac{dy}{dx}=\frac{-x}{y}$
3

Apply fraction cross-multiplication

$ydy=-xdx$
4

Integrate both sides, the left side with respect to $y$, and the right side with respect to $x$

$\int ydy=\int-xdx$
5

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{1}{2}y^2=\int-xdx$
6

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\frac{1}{2}y^2=-\int xdx$

Multiply $-1$ times $\frac{1}{2}$

$-\frac{1}{2}x^2$
7

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a constant function

$\frac{1}{2}y^2=-\frac{1}{2}x^2$
8

As the integral that we are solving is an indefinite integral, when we finish we must add the constant of integration

$\frac{1}{2}y^2=-\frac{1}{2}x^2+C_0$
9

Divide both sides of the equation by $\frac{1}{2}$

$y^2=\frac{-\frac{1}{2}x^2+C_0}{\frac{1}{2}}$

Applying the power of a power property

$y^{2\cdot \frac{1}{2}}$
10

Removing the variable's exponent

$y^{2\cdot \frac{1}{2}}=\pm \sqrt{\frac{-\frac{1}{2}x^2+C_0}{\frac{1}{2}}}$
11

Multiply $2$ times $\frac{1}{2}$

$y^{1}=\pm \sqrt{\frac{-\frac{1}{2}x^2+C_0}{\frac{1}{2}}}$
12

Any expression to the power of $1$ is equal to that same expression

$y=\pm \sqrt{\frac{-\frac{1}{2}x^2+C_0}{\frac{1}{2}}}$

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