Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. $\frac { d y } { d x } = 1 - y$

A) $y = C \ln ( 1 - y )$
B) $y = 1 x + C e ^ { - x }$
C) $y = 1 - 2 C e ^ { x }$
D) $y = C \ln ( y - 1 )$
E) $y = 1 + C e ^ { - x }$

Question

Sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. $\frac { d y } { d x } = 1 - y$

A) $y = C \ln ( 1 - y )$ 
B) $y = 1 x + C e ^ { - x }$ 
C) $y = 1 - 2 C e ^ { x }$ 
D) $y = C \ln ( y - 1 )$ 
E) $y = 1 + C e ^ { - x }$

Quizplus · Accepted Answer

From the differential equation, we can see that the slope of the solutions will always be decreasing as y increases (since the derivative is negative). This means that the solutions will approach y = 1 as x goes to infinity. Looking at the answer choices:
A) This solution doesn't satisfy the initial condition y(0) = 0 (plug in x = 0 and y = 0, and we get an undefined value)
B) This solution also doesn't satisfy the initial condition y(0) = 0 (plug in x = 0 and y = 0, and we get y = C which is not 0)
C) This solution doesn't approach y = 1 as x goes to infinity (because the exponential term will grow without bound as x goes to infinity)
D) This solution doesn't satisfy the initial condition y(0) = 0 (plug in x = 0 and y = 0, and we get an undefined value) 
E) This solution satisfies the initial condition y(0) = 0 (plug in x = 0 and y = 0 to get y = 1 + C), and it approaches y = 1 as x goes to infinity (since the exponential term goes to 0 as x goes to infinity). Therefore, the correct answer is E.

Sketch a Few Solutions of the Differential Equation on the Slope