# Differential equation solution

• Aug 26th 2010, 05:48 AM
lindah
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:"

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

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
• Aug 26th 2010, 06:47 AM
Prove It
The left hand side is a product rule expansion of $\displaystyle \frac{d}{dx}[(x^2 + 4)y]$.

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

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

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

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

Now using the boundary condition

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

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

$\displaystyle 25 = C$.

Therefore the solution is

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