Question

Math

Posted 5 months ago

```
$f^{\prime}(x)=-6 f(x)$, 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) $f(x)=2 e^{-x}$
```

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Answer from Sia

Posted 5 months ago

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.

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