dy/dt = 3y and y(0) = 2. what is t when y = 4? choose 1 answer: a t = ln(5/3) b t = 0 c t = ln(2)/3 d t =…

Answer:

C. $t = \frac{\ln 2}{3}$

Explanation:

Step1: Separate variables

We have $\frac{dy}{dt}=3y$. Rearranging gives $\frac{dy}{y}=3dt$.

Step2: Integrate both sides

Integrating $\int\frac{dy}{y}=\int 3dt$. We get $\ln|y| = 3t + C$.

Step3: Use the initial - condition

Given $y(0)=2$, substitute $t = 0$ and $y = 2$ into $\ln|y| = 3t + C$. So $\ln(2)=3\times0 + C$, then $C=\ln(2)$.

Step4: Find the general solution

The equation becomes $\ln|y| = 3t+\ln(2)$.

Step5: Solve for $t$ when $y = 4$

Substitute $y = 4$ into $\ln|y| = 3t+\ln(2)$. We have $\ln(4)=3t+\ln(2)$. Since $\ln(4)=\ln(2^2) = 2\ln(2)$, then $2\ln(2)=3t+\ln(2)$. Subtracting $\ln(2)$ from both sides gives $\ln(2)=3t$. Solving for $t$ yields $t=\frac{\ln(2)}{3}$.