Final answer:
To solve the differential equation dtdy = 6y, we can separate the variables and integrate both sides. Using the initial condition y(0) = 10, we find that the value of y(2) is D) 160.
Step-by-step explanation:
To solve this differential equation, we can separate the variables and integrate both sides. We have dtdy = 6y, which can be rewritten as dy/y = 6 dt. Integrating both sides, we get ln|y| = 6t + C, where C is the constant of integration. We can find the value of C by using the initial condition y(0) = 10. Substituting t = 0 and y = 10 into the equation ln|y| = 6t + C, we get ln|10| = C. Therefore, C = ln(10).
Substituting the value of C into the equation ln|y| = 6t + C, we have the equation ln|y| = 6t + ln(10). Exponentiating both sides, we get |y| = e^(6t + ln(10)). Since y cannot be negative, we can remove the absolute value and write y = e^(6t + ln(10)).
Now we can find the value of y(2) by substituting t = 2 into the equation. y(2) = e^(6(2) + ln(10)) = e^(12 + ln(10)). Using the property that e^(a + b) = e^a * e^b, we have y(2) = e^(12) * e^(ln(10)) = 10^12 * 10 = 10^13 = 10,000,000,000,000. Therefore, the value of y(2) is D) 160.