Find a one parameter family of solutions?

Question

Im stuck on this problem. Am I suppose to get the anti-derivative function of the differential and then plug in the initial value?

Answer 1

We are solving the differential equation. $$xy'=y$$ $$\implies$$ $$\frac{dy}{y}=\frac{dx}{x}$$ $$\implies$$ $$y=cx\quad c\in\mathbb{R}$$ We take the initial condition into account $$0=0$$ Which is a tautology, satisfied by any $c\in\mathbb{R}$. Therefore the one parameter family of solutions satisfying the initial condition $y(0)=0$ is $$y=cx\quad c\in\mathbb{R}$$

The thing to notice, if computation is to be avoided is that a free constant cannot be present in $y$, as the differentiation eliminates it, while multiplication with $x$ does not restore it. Further, one could inspect that if a polynomial is present in $y$, than differentiation would have lowered its order, but also multiply it with a constant (the order). Simply multiplying with $x$ does not undo these operations, unless the order of the polynomial is $1$. Therefore $$y=cx\quad c\in\mathbb{R}$$ are the only possible solutions.