User talk:Egm6321.f10.team5.faraone/hw2

=Problem #6=

Given

 * {| style="width:100%" border="0" align="left"

y(x)=sin^{-1}(k-15x^5)  (4) $$ 75x^4+cos(y)y'=0        (1) $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle


 * }
 * }

Find
Verify that equation 4 satisfies equation 1.

Solve
First determine $$\frac{d}{dx}y(x)$$ for equation (4).


 * $$\begin{align}

&\frac{d}{dx}sin^{-1}(x)=\frac{1}{\sqrt{1-x^2}}\\ &y'(x) = \frac{-75x^4}{\sqrt{1-(k-15x^5)^2}}\\ \end{align}$$ solve eq 1 for y'(x)
 * $$\begin{align}

&cos(sin^{-1}(x))=sin(cos{-1}(x))=\sqrt{1-x^2}\\ &75x^4+cos(y)y'=75x^4+cos(sin^{-1}(k-15x^5))y'=> y'(x)=\frac{-75x^4}{\sqrt{1-(k-15x^5)^2}}\\ \end{align}$$

$$y(x)'$$ from equation 4 is equal to $$y(x)'$$ from equation 1. Thus equations 4 satisfies equation 1.