- #1

Mayhem

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- TL;DR Summary
- Understanding this substitution for a first order homogenous equation as well as a solution to an example of such an equation.

My considers a type of differential equation $$\frac{\mathrm{d} y}{\mathrm{d} x} = f\left(\frac{y}{x} \right )$$ and proposes that it can be solved by letting ##v(x) = \frac{y}{x}## which is equivalent to ##y = xv(x)##. Then it says $$\frac{\mathrm{d} y}{\mathrm{d} x} = v + x\frac{\mathrm{d} v}{\mathrm{d} x}$$ without any further explanation. Is it correctly understand that it just let's ##\frac{\mathrm{d} y}{\mathrm{d} x} = y' = \left(xv(x) \right)' = v + x \frac{\mathrm{d} v}{\mathrm{d} x} ## ?

Next, I would like to ask if I'm understanding this right. For the DE $$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{x^2+xy+y^2}{x^2}$$ the solution is $$y = x\tan{\left(\ln(|x|) + C \right)}$$ (I got the same answer as the book). But shouldn't it be noted that this solution is only true for ##-\pi/2 < y/x < \pi/2##? The book mentions nothing of the sort in the answer key.

Next, I would like to ask if I'm understanding this right. For the DE $$\frac{\mathrm{d} y}{\mathrm{d} x} = \frac{x^2+xy+y^2}{x^2}$$ the solution is $$y = x\tan{\left(\ln(|x|) + C \right)}$$ (I got the same answer as the book). But shouldn't it be noted that this solution is only true for ##-\pi/2 < y/x < \pi/2##? The book mentions nothing of the sort in the answer key.