So as a fun project, I'm trying to work my way through Kreyzig's "Advanced Engineering Mathematics". But I've gotten to a really simple problem: $$xy' = 2y$$
where I know the solution is $x^2$ but for the life of me I can't figure out how to integrate this really simple problem properly. I keep ending up with:$$\lg(x) = 1/2\lg(2y), $$
but I don't think that's right. Help?
$\endgroup$ 21 Answer
$\begingroup$THis is a problem where we may use separation of variables. That is we begin by moving the $x$ terms to the right side and $y$ terms to the left so that we have $$\frac{y'}{y} = \frac{2}{x}.$$
We then integrate both sides and have $$\int \frac{1}{y} dy = \int \frac{2}{x} dx$$ which results in $$ln(y) =2\ln(x) + C.$$ Solving for $y$ by exponentiating both sides yields $$y=e^{\ln(x^2) + c} = e^{\ln(x^2)}e^c = Ax^2$$ where $A = e^c$.
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