Consider the differential equation dy/dx= e^x-1/2y. If y = 4 when x = 0 what is a value of y when x = 1?

A. √(e + 14)

Step-by-step explanation:

separating the variables, we have

2ydy = ( - 1)dx

integrating, we have

∫ydy = ∫( - 1)dx

∫ydy = ∫dx - ∫1dx + C

2y²/2 = - x + C

y² = - x + C

when x = 0, y = 4

So,

y² = - x + C

4² = - 0 + C

16 = 1 + C

16 = 1 + C

C = 16 - 1

C = 15

So,

y² = - x + 15

when x = 1, we find y

y² = - 1 + 15

y² = e + 14

y = √(e + 14)

answer is 8x+24

area of vanessa's patio = length × width

given, length = (3x² + 5x + 10)

width = (x² - 3x - 1)

area of vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)

= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)

= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10

= 3x⁴ - 4x³ - 8x² - 35x - 10

hence, expression of vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)