Answer to Question #230920 in Civil and Environmental Engineering for syra

Question

Find the equation of the family of curve having the given slope, and the equation of the family which passes through the given point.

m= 2x⁴/x; (0,5)

Note: the slope m = dy/dx

Answer 1

Considering the following equation of the slope

$\frac{dy}{dx}= \frac{2x^4}{x}= 2x^3$

Separating the variables

$dy = 2x^3 dx\\ \int dy = \int 2x^3 dx\\ y= \frac{2x^4}{4}+C\\ y= \frac{x^4}{2}+C\\$

Finding the value of C

$y= \frac{x^4}{2}+C\\ 5= \frac{0^4}{2}+C\\ C=5$

The equation of the curve is given as

$y= \frac{x^4}{2}+5\\$