Question

This is a Differential Equations problem, I only need help on part 1 from question five. I need steps as well, thank you.

Answer 1

a)
`y(t) = √(2t + 1 )`

b) 1.5 hours after thickness will be 2 inches.

Explanation:

`(dy)/(dt) = (1)/(y) \\ \\ ydy = dt \\ \\ integrating \: both \: sides \\ \\ \int y \: dy = \int 1 \: dt \\ \\ \frac{ {y}^(2) }{2} = t + c \\ \\ {y}^(2) = 2t + 2c \\ \\ y (t)= √(2t + 2c) ...(1) \\ \\ a) \: \: \: plug \: t = 0 \: in \: (1) \\ \\ y (0)= √(2(0) +2 c) \\ \\ y (0)= √(0 +2 c) \\ \\ 1 = √(2c) \: \: ( \because \: y (0)=1) \\ \\ 2c = 1 \: \implies \: c = (1)/(2) \\ \\ plug \: c = (1)/(2) \: in \: (1) \\ \\ y(t) = \sqrt{2t + 2 * (1)/(2) } \\ \\ \huge \: \red {y(t) = √(2t + 1 ) } \\ \\b) \: \: plug \: y(t) = 2 \: in \: above \: equation \\ \\ 2 = √(2t + 1) \\ \\ 4 = 2t + 1 \: \\ (squaring \: both \: sides) \\ \\ 4 - 1 = 2t \\ \\ 2t = 3 \\ \\ t = (3)/(2) \\ \\ t = 1.5 \: hours \\`