Questions: The equation is: (1/2)f''+ 4 (1/t)f'+ (k/t^2)f = 200, where t is the time in minutes, and k is a positive constant. (e) Let f(t) = t^n, where n is a constant. Find the value of n.

The equation is: (1/2)f''+ 4 (1/t)f'+ (k/t^2)f = 200, where t is the time in minutes, and k is a positive constant.

(e) Let f(t) = t^n, where n is a constant. Find the value of n.
Transcript text: The equation is: $\frac{1}{2}f''+ 4 \frac{1}{t}f'+ \frac{k}{t^2}f = 200$, where $t$ is the time in minutes, and $k$ is a positive constant. (e) Let $f(t) = t^n$, where $n$ is a constant. Find the value of $n$.
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Solution

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Solution Steps

Step 1: Substitute f(t)=tn f(t) = t^n into the equation

Given the function f(t)=tn f(t) = t^n , we need to find its first and second derivatives:

f(t)=ddt(tn)=ntn1 f'(t) = \frac{d}{dt}(t^n) = n t^{n-1}

f(t)=d2dt2(tn)=n(n1)tn2 f''(t) = \frac{d^2}{dt^2}(t^n) = n(n-1) t^{n-2}

Substitute these into the given differential equation:

12f+41tf+kt2f=200 \frac{1}{2}f'' + 4 \frac{1}{t}f' + \frac{k}{t^2}f = 200

12n(n1)tn2+41tntn1+kt2tn=200 \frac{1}{2} \cdot n(n-1) t^{n-2} + 4 \cdot \frac{1}{t} \cdot n t^{n-1} + \frac{k}{t^2} \cdot t^n = 200

Step 2: Simplify the equation

Simplify each term:

  1. 12n(n1)tn2\frac{1}{2} n(n-1) t^{n-2} remains as is.
  2. 41tntn1=4ntn24 \cdot \frac{1}{t} \cdot n t^{n-1} = 4n t^{n-2}.
  3. kt2tn=ktn2\frac{k}{t^2} \cdot t^n = k t^{n-2}.

Combine these terms:

12n(n1)tn2+4ntn2+ktn2=200 \frac{1}{2} n(n-1) t^{n-2} + 4n t^{n-2} + k t^{n-2} = 200

Factor out tn2 t^{n-2} :

(12n(n1)+4n+k)tn2=200 \left(\frac{1}{2} n(n-1) + 4n + k\right) t^{n-2} = 200

Step 3: Solve for n n

For the equation to hold for all t t , the power of t t on both sides must match. Since the right side is a constant (200), the power of t t on the left must be zero:

n2=0    n=2 n-2 = 0 \implies n = 2

Substitute n=2 n = 2 back into the coefficient equation:

122(21)+42+k=200 \frac{1}{2} \cdot 2(2-1) + 4 \cdot 2 + k = 200

1+8+k=200 1 + 8 + k = 200

k=2009=191 k = 200 - 9 = 191

Final Answer

The value of n n is 2\boxed{2}.

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