Solve $\frac{dy}{dx} (xy)+ 4x^2 + y^2 = 0$ using the substitution $w=\frac yx$ Solve $\frac{dy}{dx} (xy)+ 4x^2 + y^2 = 0$ using the substitution $w=\frac yx$
I have done most of this question but i'm unsure how to write the solution. This is what I have so far:
$$\frac{dy}{dx} + 4(x/y) + \frac yx = 0$$
$$\frac{dy}{dx} + \frac4w + w=0$$
As $w=\dfrac yx$, it follows that $y=xw$ so $\dfrac{dy}{dx} = w$.
so we now have:
$$w +\dfrac4w + w = 0$$
$$2w^2 + 4 = 0$$
Which gives the solution  $\pm\sqrt{2i}$
This is now the point where I get confused. 
Is this next part correct?
$$w = C_1 \cos(x\sqrt2) + C_2 \sin(x\sqrt2)$$
Therefore:
$$y =  x[ C_1\cos(x\sqrt2) + C_2\sin(x\sqrt2)]$$
Where $C_1$ and $C_2$ are constants. I have a feeling this isn't correct?
 A: Your feeling that your solution isn't correct is indeed right. Everything is correct until the following line:

As $w=\dfrac yx$, it follows that $y=xw$ so $\color{red}{\dfrac{dy}{dx} = w}$.

This answer will rectify this by discussing how to solve a specific type of ordinary differential equation where we in general can use the substitution $w=\dfrac{y}{x}$. You should then be able to apply this method for your specific ODE.

In general, a homogeneous differential equation is one which can be written in the form:
$$\frac{dy}{dx}=F\left(\frac{y}{x}\right) \tag{1}$$
Your equation can clearly be written in the above form.
Using the proposed substitution, $w=\dfrac{y}{x}$, which is $y=x\cdot w$ as you correctly mentioned, it follows from the product rule that:
$$\frac{dy}{dx}=1\cdot w+x\cdot \frac{dw}{dx}$$
Which is clearly not $\dfrac{dy}{dx}=w$, as pointed out by @MatthewLeingang.
Therefore, you can write $(1)$ as:
$$w+x\frac{dw}{dx}=F(w) \tag{2}$$
Which is a separable ODE, which can be solved by rewriting $(2)$:
$$\frac{1}{F(w)-w}\cdot \frac{dw}{dx}=\frac{1}{x}\implies \int\frac{dw}{F(w)-w}=\int\frac{dx}{x}$$
