Use as a integrating factor to find the general solution of the differential equation .

Question

Use  as a integrating factor to find the general solution of the differential equation  .

Quizplus · Accepted Answer

We need to find an integrating factor, say $u$, for the differential equation $y' + 3y = 6$. An integrating factor $u$ for this equation satisfies the differential equation $u' y + u y' = 0$, where $y' + 3y = 6$ is the original differential equation.

So we have $u' y + u y' = (uy)' = 0$. Thus, $uy = C$, where $C$ is a constant.

Using the given integrating factor $11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11$, we have

$$11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11 \cdot y' + 11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11 \cdot 3y = 11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11 \cdot 6$$

We notice that the left side of this equation can be written as the derivative of $11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11 \cdot y$, thus giving us

$$(11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11 \cdot y)' = 11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11 \cdot 6$$

Integrating both sides, we get

$$11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11 \cdot y = \frac{11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11}{6} \cdot 6t + C$$

where $C$ is a constant. Simplifying, we have

$$y = t + \frac{C}{11eaa595\_6521\_b7fc\_a696\_ed6166885eec\_TB4584\_11\_TB4584\_11}$$

which is the general solution.

We observe that option C is the given integrating factor, so the answer is (C).

Use as a Integrating Factor to Find the General