Question

Given y'(x) = alt="\sqrt{-4y(x) + 28}" align="absmiddle" class="latex-formula"> and y(-2)=-2, find y(x)?

Answer 1

I saw this question the first time around missing those parentheses around the $x$, which made the ODE much, much more difficult to solve. This is doable, especially because this ODE is separable.

$\dfrac{\mathrm dy}{\mathrm dx}=\sqrt{-4y+28}$
$\dfrac{\mathrm dy}{2\sqrt{7-y}}=\mathrm dx$
$\displaystyle\frac12\int\frac{\mathrm dy}{\sqrt{7-y}}=\int\mathrm dx$
$-\sqrt{7-y}=x+C$
$\sqrt{7-y}=-x+C$
$7-y=(C-x)^2$
$y=7-(C-x)^2$

Given that $y(-2)=-2$, we have

$-2=7-(C+2)^2$
$\implies C=-5,C=1$

So there are two possible particular solutions:

$\begin{cases}y=-x^2-10x-18\\y=-x^2+2x+6\end{cases}$