Which of the following is the solution to the differentiable equation dy/dx=e^y+x with the initial condtion y(0)=-ln(4)

y = -ln(-e^x + 5)

Step-by-step explanation:

I assume you mean:

dy/dx = e^(y + x)

Use exponent properties:

dy/dx = (e^y) (e^x)

Separate the variables:

e^(-y) dy = e^x dx

Integrate:

-e^(-y) = e^x + C

Solve for y:

e^(-y) = -e^x − C

-y = ln(-e^x − C)

y = -ln(-e^x − C)

Use initial condition to solve for C:

-ln 4 = -ln(-e^0 − C)

4 = -1 − C

C = -5

Therefore, the equation is:

y = -ln(-e^x + 5)

the correct answer is option c.   3

step-by-step explanation:

points to remember

equation of the line with coordinates (x₁, y₁) and (x₂, y₂) is given by,

slope m = (y₂ - y₁)/(x₂ - x₁)

from the figure we can see a line with two coordinates given

to find the slope of given line

here   (x₁, y₁)   = (-1, -1 ) and (x₂, y₂) = (1, 5)

slope m = (y₂ - y₁)/(x₂ - x₁)

  = (5 - -1)/(1 - -1)

  = (5 + 1)/(1 + 1)

  = 6/2 = 3

therefore the correct answer is option c.   3