Let S(t) be a solution to the differential equation
t
dS
dt − S = 3
with the initial value S(1) = 1. Which of the following is the value of S(2)?
A. 0 B. 2 C. π D. 5 E. −5
Huh?
try typing it so it makes sense. copy/paste don't do well here
dS/dt ... ?
To find the value of S(2), we need to solve the given differential equation and apply the initial condition S(1) = 1.
The differential equation is:
t(dS/dt) - S = 3
To solve this, we can use the method of separation of variables. Rearrange the equation to isolate dS on one side:
dS = (S + 3)/t dt
Now, we can separate the variables and integrate both sides:
∫ dS / (S + 3) = ∫ dt / t
Integrating the left side:
(ln |S + 3|) = ln |t| + C1
where C1 is the constant of integration.
Taking the exponent of both sides:
|S + 3| = e^(ln|t| + C1) = e^(ln|t|) * e^(C1) = C*t, (where C = e^(C1))
Considering the absolute value, we have two cases:
Case 1: S + 3 > 0
S + 3 = C * t
Case 2: S + 3 < 0
-(S + 3) = C * t
Now, let's apply the initial condition S(1) = 1:
Case 1: S + 3 > 0
S(1) + 3 = C * 1
1 + 3 = C
C = 4
Case 2: S + 3 < 0
-(S(1) + 3) = C * 1
-(1 + 3) = C
C = -4
So, we have two possible solutions:
Case 1: S + 3 = 4t
Case 2: -(S + 3) = -4t
Now, let's check which solution satisfies the condition S(2) and figure out the corresponding value.
Case 1: S + 3 = 4t
Substituting t = 2, we get:
S(2) + 3 = 4*2
S(2) + 3 = 8
S(2) = 8 - 3 = 5
Case 2: -(S + 3) = -4t
Substituting t = 2, we get:
-(S(2) + 3) = -4*2
-S(2) - 3 = -8
S(2) = -8 + 3 = -5
So, the possible values for S(2) are 5 and -5.
Therefore, the correct answer is E. -5.