Given dy/dt=4y and y(2)=200. Find y(9)
dy/dt = 4y
dy/y = 4 dt
lny = 4t + C
y = ce^(4t)
y(2) = 200 means
c*e^8 = 200
c = 200e^-8
y = 200e^-8 * e^(4t) = 200e^(4t-8)
y(9) = 200e^28
Well, first of all, let's address the fact that we have a "dy/dt" equation. It kind of sounds like a rapper's stage name, don't you think? "Yo, coming to the stage, it's Dy/dt!" Okay, okay, I'll stop with the silly jokes and get serious now.
To solve this differential equation, we can separate the variables and integrate both sides. Let's start by dividing both sides by y:
dy/y = 4 dt
Now, let's integrate both sides. The integral of 1/y with respect to y is natural logarithm of the absolute value of y (ln|y|), and the integral of 4 dt is just 4t. So we have:
ln|y| = 4t + C
Now, let's use the initial condition y(2) = 200 to find the value of the constant C. Plugging in t = 2 and y = 200:
ln|200| = 4(2) + C
ln(200) = 8 + C
To find C, we can subtract 8 from both sides:
C = ln(200) - 8
Now, we have the specific equation for our problem:
ln|y| = 4t + ln(200) - 8
Finally, we can solve for y(9). Plugging in t = 9 into our equation:
ln|y(9)| = 4(9) + ln(200) - 8
ln|y(9)| = 36 + ln(200) - 8
To get rid of the natural logarithm, we can exponentiate both sides:
|y(9)| = e^(36 + ln(200) - 8)
Now, taking the absolute value of a number doesn't change the number itself, so we can drop the absolute value signs:
y(9) = e^(36 + ln(200) - 8)
And there you have it! That's the value of y(9) using the given differential equation and initial condition. But wait, don't think I forgot about clowning around! Here's a joke for you: Why did the mathematician name his dog "Cauchy"? Because he left a residue on the sofa! Keep laughing and learning, my friend!
To find y(9), we need to solve the given differential equation dy/dt = 4y and use the initial condition y(2) = 200.
Step 1: Separate the variables:
dy/y = 4dt
Step 2: Integrate both sides:
∫(dy/y) = ∫4dt
Step 3: Evaluate the integrals:
ln|y| = 4t + C1
(ln|y| is the natural logarithm of the absolute value of y, and C1 is the constant of integration)
Step 4: Exponentiate both sides:
|y| = e^(4t + C1)
Step 5: Apply the initial condition y(2) = 200:
When t = 2, y = 200:
|200| = e^(4(2) + C1)
200 = e^(8 + C1)
Step 6: Solve for C1:
Take the natural logarithm of both sides to isolate C1:
ln(200) = 8 + C1
Subtract 8 from both sides:
C1 = ln(200) - 8
Step 7: Substitute C1 back into the equation for y:
|y| = e^(4t + [ln(200) - 8])
Step 8: Simplify the absolute value:
Since e^(4t + [ln(200) - 8]) is always positive, we can remove the absolute value sign:
y = e^(4t + [ln(200) - 8])
Step 9: Evaluate y(9):
When t = 9:
y = e^(4(9) + [ln(200) - 8])
y = e^(36 + [ln(200) - 8])
y ≈ 2615.83 (rounded to two decimal places)
Therefore, y(9) is approximately 2615.83.
To solve the differential equation dy/dt = 4y, we can use separation of variables.
Step 1: Separate the variables by moving dy to one side and dt to the other side.
dy/y = 4 dt
Step 2: Integrate both sides with respect to their respective variables.
∫(1/y) dy = ∫4 dt
Step 3: Evaluate the integrals.
ln|y| = 4t + C
Step 4: Solve for y by exponentiating both sides.
|y| = e^(4t+C)
Step 5: Remove the absolute value by considering two cases.
Case 1: y > 0
y = e^(4t+C)
Case 2: y < 0
y = -e^(4t+C)
Since y(2) = 200, we substitute t = 2 and solve for C.
200 = e^(4*2+C)
200 = e^(8+C)
To solve for C, take the natural logarithm of both sides.
ln(200) = ln(e^(8+C))
ln(200) = 8 + C
Subtract 8 from both sides to isolate C.
C = ln(200) - 8
Now we can determine y(9) by substituting t = 9 into the equations for y:
For y > 0:
y(9) = e^(4*9+(ln(200)-8))
= e^(36+ln(200)-8)
For y < 0:
y(9) = -e^(4*9+(ln(200)-8))
= -e^(36+ln(200)-8)
Using a calculator, we can simplify the expressions and compute the numerical value of y(9) for both cases.