# Thread: No idea how to find this numerical value from this first order differential equation

1. ## No idea how to find this numerical value from this first order differential equation

Here is the question (and Answer under "Differential equation solution"):
"Find the solution of the differential equation that satisfies the given initial condition."
dy&#x2f;dx &#x3d; x&#x2f;y, y&#x28;0&#x29; &#x3d; -3 - Wolfram|Alpha

I'm ok when it involves only variables but when it gives me y(0) = -3, it throws me off.
My work:

dy/dx = x/y
integral of ydy = integral of xdx
1/2 * y^2 = 1/2 * x^2
y = x

That makes no sense.

Can someone please explain to me what the question is really asking as well as help me solve it? I get confused because it gives me numbers and the answer is in variables.

2. you forget the integration constant, i mean, you must to have: $\displaystyle y^2=x^2+k$ and whit y(0)=-3 you can obtain value of $\displaystyle k$

3. $\displaystyle \int y \, dy = \int x \, dx$

$\displaystyle \implies \tfrac{1}{2}y^2+C_1 = \tfrac{1}{2}x^2+C_2$

You can just combine the two constants and get

$\displaystyle \implies \tfrac{1}{2}y^2 = \tfrac{1}{2}x^2+C$

Then you can simplify further and use the initial condition to solve for the constant.

(This is basically repeating what Nacho said, but I just wanted to make it clear why there is only one constant instead of two.)

4. Ok so I get

y = +/- sqrt(x^2 + 9)