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I am stuck trying to solve for the below ODE,
$$ \dfrac{d y}{dx}=\dfrac{y}{x}+1 $$
it would be trivial to solve if it did not have the one at the end since I could use separation of variables. I tried to use a change of variables $ y = \xi -x$ but that did not get me anywhere. Does a simple solution to this ODE exist?
2 Answers
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HINT:
Set $\displaystyle\frac yx=v$
$\displaystyle\implies y=vx\implies\frac{dy}{dx}=v+x\frac{dv}{dx}$
Reference : Homogeneous Ordinary Differential Equation
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$$ \dfrac{d y}{dx}=\dfrac{y}{x}+1\quad\Longrightarrow\quad \frac{1}{x}\dfrac{d y}{dx}-\frac{1}{x^2}y=\frac{1}{x}\quad\Longrightarrow\quad \left(\frac{y}{x}\right)'-(\ln x)'=0\quad\Longrightarrow\quad \frac{y}{x}=\ln x+c, $$ for some $c$ constant.
