$\frac{d y}{d t}=y+1 \text { and } y(0)=3 \text {. }$ $\newline$ What is $t$ when $y=1$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $t=0$ $\newline$ (B) $t=\ln 2$ $\newline$ (C) $t=-\ln 2$ $\newline$ (D) $t=1$ $\newline$ (E) $t=\ln 4$

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Euler's method

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\frac{d y}{d t}=y+1 \text { and } y(0)=3 \text {. }

\newline

What is

t

when

y=1

\newline

Choose

1

answer:

\newline

(A)

t=0

\newline

(B)

t=\ln 2

\newline

(C)

t=-\ln 2

\newline

(D)

t=1

\newline

(E)

t=\ln 4

Solve Differential Equation: First, we need to solve the differential equation $\frac{dy}{dt} = y + 1$ with the initial condition $y(0) = 3$ . We can separate variables and integrate.
Separate Variables: Separate the variables: (\frac{dy}{y+$1$} = dt)\.
Integrate Both Sides: Integrate both sides: \(\int\frac{1}{y+1}\, dy = \int dt.
Find Constant of Integration: The integral of $\frac{1}{y+1}$ with respect to $y$ is $\ln|y+1|$ , and the integral of $dt$ is $t$ . So we have $\ln|y+1| = t + C$ , where $C$ is the constant of integration.
Exponentiate to Solve for y: Using the initial condition $y(0) = 3$ , we can find $C$ . Plugging in the values, we get $\ln|3+1| = 0 + C$ , which simplifies to $\ln(4) = C$ .
Substitute $y = 1$ : Now we have the particular solution $\ln|y+1| = t + \ln(4)$ . We can solve for $y$ by exponentiating both sides to get rid of the natural logarithm.
Solve for $e^t$ : Exponentiate both sides: $e^{\ln|y+1|} = e^{t + \ln(4)}$ , which simplifies to $|y+1| = e^t \cdot 4$ .
Take Natural Logarithm: Since we are looking for when $y = 1$ , we substitute $1$ into the equation: $|1+1| = e^t \times 4$ , which simplifies to $2 = e^t \times 4$ .
Final Solution: Divide both sides by $4$ to solve for $e^t$ : $e^t = \frac{2}{4}$ , which simplifies to $e^t = \frac{1}{2}$ .
Final Solution: Divide both sides by $4$ to solve for $e^t$ : $e^t = \frac{2}{4}$ , which simplifies to $e^t = \frac{1}{2}$ .Take the natural logarithm of both sides to solve for $t$ : $\ln(e^t) = \ln(\frac{1}{2})$ , which simplifies to $t = \ln(\frac{1}{2})$ .
Final Solution: Divide both sides by $4$ to solve for $e^t$ : $e^t = \frac{2}{4}$ , which simplifies to $e^t = \frac{1}{2}$ .Take the natural logarithm of both sides to solve for $t$ : $\ln(e^t) = \ln(\frac{1}{2})$ , which simplifies to $t = \ln(\frac{1}{2})$ .We know that $\ln(\frac{1}{2})$ is the same as $-\ln(2)$ , so $t = -\ln(2)$ .