find solution to differential equation given the initial conditions DQ/dt=Q/5
Q= 50 when t= 0
i don't understand this one because of the division sign.
dQ/Q = dt/5
ln Q = t/5 + C
Q = C' exp (t/5)
C and C' are arbitiry constants that you can determine from the initial condition
Q = 50 exp (t/5)
To find the solution to the differential equation DQ/dt = Q/5 with the given initial condition Q = 50 when t = 0, we can separate the variables and solve it using the method of separation of variables.
First, let's rearrange the equation to separate the variables:
DQ = (Q/5) dt
Now, let's move the dt term to the right side:
(1/Q) dQ = (1/5) dt
Next, integrate both sides of the equation:
∫(1/Q) dQ = ∫(1/5) dt
To integrate (1/Q) with respect to Q, we can use the natural logarithm:
ln|Q| = (1/5)t + C
where C is the constant of integration.
Now, let's solve for Q by taking the exponential of both sides:
|Q| = e^[(1/5)t + C]
Since we need a specific solution, let's consider the initial condition Q = 50 when t = 0:
|50| = e^[(1/5)(0) + C]
50 = e^(C)
To determine the value of C, we need to know the sign of Q. Since the absolute value signs are present, both positive and negative solutions are valid.
Case 1: Q > 0
If Q is positive, then |Q| = Q, and the equation becomes:
50 = e^(C)
Taking the natural logarithm of both sides:
ln(50) = C
Case 2: Q < 0
If Q is negative, then |Q| = -Q, and the equation becomes:
50 = -e^(C)
Taking the natural logarithm of both sides:
ln(-50) = C
Therefore, we have two possible solutions:
Case 1: Q > 0
Q = e^(ln(50)) * e^[(1/5)t] = 50 * e^[(1/5)t]
Case 2: Q < 0
Q = -e^(ln(-50)) * e^[(1/5)t] = -50 * e^[(1/5)t]
In summary, the general solution to the given differential equation DQ/dt = Q/5 with the initial condition Q = 50 when t = 0 is given by:
Q = 50 * e^[(1/5)t]
or
Q = -50 * e^[(1/5)t]