# Math Help - Differential equation solution

1. ## Differential equation solution

Hello,
I don't understand what the following question is asking - may I ask for someone to elaborate?:

"By considering the differential of z(x)=f(x)y(x), find the solution of the following differential equation:"

$(x^2+4)dy/dx + 2xy =0$ where $y(1) =5$

I've looked up solving differential equations, but most involve integration, which we have not touched on in lectures yet.
Or am I completely off track?

Thank you in advance for any feedback

2. The left hand side is a product rule expansion of $\frac{d}{dx}[(x^2 + 4)y]$.

So $\frac{d}{dx}[(x^2 + 4)y] = 0$

$(x^2 + 4)y = \int{0\,dx}$

$(x^2 + 4)y = C$

$y = \frac{C}{x^2 + 4}$.

Now using the boundary condition

$5 = \frac{C}{1^2 + 4}$

$5 = \frac{C}{5}$

$25 = C$.

Therefore the solution is

$y = \frac{25}{x^2 + 4}$.