Question

A curve is such that dy∕dx=√ (2x+5) and (2,5) is a point on the curve. What is the equation of the curve?

Answer 1

Given

$\frac{d y }{d x}=\sqrt{( 2 x+5)}$

Separate variables

$\begin{aligned} dy&= \sqrt{2x+5}dx\\ y&=\int \sqrt{2x+5}dx\\ &=\frac{1}{2}\int 2\sqrt{2x+5}dx\\ &=\frac{1}{2}\frac{2}{3} (2x+5)^{3/2}+c\\ &= \frac{1}{3}(2x+5)^{3/2}+c \end{aligned}$

Since the curve passes through (2,5) , then

$\begin{aligned} 5&= \frac{1}{3}(4+5)^{3/2}+c\\ 5&= 9+c\\ c&=-4 \end{aligned}$

Hence

$y= \frac{1}{3}(2x+5)^{3/2}-4$