(dy)/(dt)=2t+3" and "y(1)=6 What is t when y=0 ? Choose all answers that apply: A t=-1 B t=-3 C t=0 D t=-4 ㅌ t=1 F t=-2

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(dy)/(dt)=2t+3" and "y(1)=6 What is  t when  y=0 ? Choose all answers that apply: A  t=-1 B  t=-3 C  t=0 D  t=-4 ㅌ  t=1 F  t=-2

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Integrate Equation: Given the differential equation (dy)/(dt) = 2t + 3 and the initial condition y(1) = 6, we want to find the value of t when y = 0. To do this, we need to integrate the differential equation to find the general solution for y(t).Find General Solution: Integrate the right-hand side of the equation with respect to t to find y(t). The integral of 2t with respect to t is t^2, and the integral of 3 with respect to t is 3t. So, the integral of (dy)/(dt) = 2t + 3 is y(t) = t^2 + 3t + C, where C is the constant of integration.Use Initial Condition: Use the initial condition y(1) = 6 to find the value of the constant C. Plugging in the values, we get 6 = 1^2 + 3(1) + C, which simplifies to 6 = 1 + 3 + C. Therefore, C = 6 - 4 = 2.Write Particular Solution: Now that we have the constant, we can write the particular solution for y(t) as y(t) = t^2 + 3t + 2.Set Equation Equal to Zero: To find the value of t when y = 0, we set the equation y(t) = t^2 + 3t + 2 equal to 0 and solve for t. This gives us the quadratic equation t^2 + 3t + 2 = 0.Factor Quadratic Equation: Factor the quadratic equation to find the values of t. The equation t^2 + 3t + 2 = 0 factors into (t + 1)(t + 2) = 0.Solve for t: Set each factor equal to zero and solve for t. This gives us t + 1 = 0 and t + 2 = 0, which means t = -1 and t = -2 are the solutions.Check Answer Choices: We can now check the answer choices to see which ones apply. The correct answers are A t = -1 and F t = -2.

$\frac{d y}{d t}=2 t+3 \text { and } y(1)=6$ $\newline$ What is $t$ when $y=0$ ? $\newline$ Choose all answers that apply: $\newline$ (A) $t=-1$ $\newline$ (B) $t=-3$ $\newline$ (C) $t=0$ $\newline$ (D) $t=-4$ $\newline$ (E) $t=1$ $\newline$ (F) $t=-2$

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