$f^{\prime}(x)=-6 f(x)$, and $f(2)=1$.
Solve the equation.

Choose 1 answer:
(A) $f(x)=-12 e^{-6 x}$
(B) $f(x)=e^{-6 x}$
(C) $f(x)=e^{12-6 x}$
(D) $f(x)=2 e^{-x}$

Question

$f^{\prime}(x)=-6 f(x)$, and $f(2)=1$.
Solve the equation.

Choose 1 answer:
(A) $f(x)=-12 e^{-6 x}$
(B) $f(x)=e^{-6 x}$
(C) $f(x)=e^{12-6 x}$
(D) $f(x)=2 e^{-x}$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Solve the differential equation $f'(x) = -6f(x)$ using separation of variables. ‖ ‖ step 2 ⋮ The general solution is $f(x) = c_1 e^{-6x}$, where $c_1$ is a constant. ‖ ‖ step 3 ⋮ Apply the initial condition $f(2) = 1$ to find the value of $c_1$. ‖ ‖ step 4 ⋮ Substitute $x = 2$ into the general solution to get $1 = c_1 e^{-12}$. ‖ ‖ step 5 ⋮ Solve for $c_1$ to get $c_1 = e^{12}$. ‖ ‖ step 6 ⋮ Substitute $c_1$ back into the general solution to get the specific solution $f(x) = e^{12-6x}$. ‖ ∻Answer∻ ⚹ $f(x) = e^{12-6x}$ ⚹ ∻Key Concept∻ ⚹Solving first-order linear ordinary differential equations with initial conditions⚹ ∻Explanation∻ ⚹The solution to the differential equation is found by integrating both sides and then applying the given initial condition to solve for the constant of integration.⚹◊What is the general solution for $f(x)$ given the differential equation $f'(x)=-6f(x)$ and initial condition $f(2)=1$?⍭ Generate me a similar question◊

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