How can the function dy/dx=x^y be integrated?

Thursday, October 18, 2012

Given the function:

$\displaystyle\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}={{x}}^{{y}}$

It can be integrated as:

$\displaystyle\int{\left({\left.{d}{y}\right.}\right)}=\int{{x}}^{{y}}{\left.{d}{x}\right.}$

We can use the power rule of integration to solve this case. The power rule is

$\displaystyle\int{{x}}^{{n}}=\frac{{{{x}}^{{{n}+{1}}}}}{{{n}+{1}}}$

where, n is a non-zero number.

Assuming that y $\displaystyle\ne$ 0, the given integral can be solved as:

$\displaystyle\int{{x}}^{{y}}{\left.{d}{x}\right.}=\frac{{{x}}^{{{y}+{1}}}}{{{y}+{1}}}+{C}$

Thus, the power rule helps us in getting the solution to the given integral.

We can also calculate the value of C (the constant of integration), if we are provided with some limits, or some values of y, given values of x.

For example, let us say when x =1, y =1.

In that case, y = 1 = x^(y+1) / (y+1) + C = 1^2 / (1 + 1) + C

or, C = 1/2.

Hope this helps.

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