Given dy/dt=4y and y(2)=450. Find y(8).
dy/dt = 4y
dy/y = 4 dt
lny = 4t+ln(c)
y = c*e^(4t)
y(2)=450, so
c*e^2 = 450
c = 450/e^2 ≈ 60.9
y = 60.9 e^(4t)
so, now find y(8)
I keep getting some huge number when I do 60.9(e^(4*8)) and I know that's not right..
Let me recheck Steves response
lny = 4t+ln(c)
y = ke^4t
y(2)=450, so
c*e^8 = 450 Here is where I change to my work.
c = 450/e^8 ≈ 2981
y = 2981 e^(4t)
y(8)=2981e^16=2.64e10
Good call. I forgot the 4t
But 450/e^8 = 0.15
The answer is still big, though.
To solve the differential equation, dy/dt = 4y, we can separate the variables and integrate both sides. Here's how you can solve it step by step:
Step 1: Separate variables
dy/y = 4dt
Step 2: Integrate both sides
∫(1/y) dy = ∫(4) dt
Step 3: Evaluate the integrals
ln|y| = 4t + C
Step 4: Solve for y
Taking the exponential of both sides gives us:
e^(ln|y|) = e^(4t + C)
|y| = e^(4t) * e^C
Step 5: Determine the constant of integration
Since we have an initial condition, y(2) = 450, we can substitute the values into our equation and solve for the constant C.
When t = 2:
|450| = e^(4 * 2) * e^C
450 = e^(8 + C)
Taking the natural log of both sides:
ln(450) = 8 + C
C = ln(450) - 8
So, the equation becomes:
|y| = e^(4t) * e^(ln(450) - 8)
Step 6: Solve for y(8)
Now that we have the equation, we can substitute t = 8 into it and solve for y(8).
When t = 8:
|y(8)| = e^(4 * 8) * e^(ln(450) - 8)
y(8) = e^(32) * (450 / e^8)
Evaluating this expression, we can find the value of y(8).