The particular solution of the differential equation dy/dt= 2y for which y(0) = 60 is
a) y= 60e^2t
b) y= 60e^0.5t
c) y= 59 + e^t
d) y= 30e^t
now, I know you can do this ...
dy/y = 2 dt
ln y = 2t + ln c
y = c*e^(2t)
...
I don't understand how to get the solution to this problem
I assume you get to my last step ok.
y = c*e^(2x)
y(0) = 60, s0
c*e^0 = 60
c = 60
y = 60e^(2x)
Looks like you need to do a bunch of DE problems to get the technique down. Lots of online example, and I'm sure your text has some worked out as well.
To solve this differential equation, we can separate the variables and integrate both sides.
The given differential equation is: dy/dt = 2y
Step 1: Separating the variables
Divide both sides by y to isolate dy on the left side and dt on the right side:
(1/y) dy = 2 dt
Step 2: Integrating both sides
Integrate both sides with respect to their respective variables:
∫(1/y) dy = ∫2 dt
Step 3: Evaluating the integrals
The integral of (1/y) dy can be evaluated as ln|y|, and the integral of 2 dt is 2t. Adding the constant of integration, C, gives:
ln|y| = 2t + C
Step 4: Solving for y
To solve for y, we need to eliminate the natural logarithm. Taking the inverse of the natural logarithm, we can rewrite the equation as:
|y| = e^(2t + C)
Now, let's consider the initial condition given: y(0) = 60. We can substitute this into the equation to find the value of the constant, C.
When t = 0, the equation becomes:
|60| = e^(2(0) + C)
60 = e^C
Since e^C is always positive, we can remove the absolute value signs:
60 = e^C
Step 5: Final solution
Substituting the value of C back into the equation, we get:
|y| = e^(2t + ln(60))
Now, since the absolute value of y can take on any value, we have two possibilities for the equation:
1) y = e^(2t + ln(60))
2) y = -e^(2t + ln(60))
However, the given initial condition requires y(0) = 60, which means we should choose the positive solution:
y = e^(2t + ln(60))
Now let's simplify this equation:
y = e^ln(60) * e^(2t)
y = 60 * e^(2t)
Therefore, the particular solution of the given differential equation with the initial condition y(0) = 60 is:
y = 60 * e^(2t)
So, the correct answer is option a) y = 60e^(2t).