Диференциално уравнение

[555]

y`= y/x + e^(–y/x)

[Nathi123]

y'= [tex]\frac{y}{x} + e^{-\frac{y}{x}} , \frac{y}{x} = t \Rightarrow y = tx
\Rightarrow y ' = t'x + t \Rightarrow t'x + t = t + e^{-t} \Leftrightarrow t'x = e^{-t} \Leftrightarrow[/tex]
[tex]\frac{xdt}{dx} = e^{-t}
\Rightarrow \frac{dt}{e^{-t} } = \frac{dx}{x} , x\ne 0\Rightarrow e^{t}dt = \frac{dx}{x} \Rightarrow \int e^{t} dt = \int\frac{dx}{x}\Leftrightarrow e^{t} = ln|x| +ln C\Rightarrow e^{t} =lnC|x|[/tex] и
[tex]e^{\frac{y}{x}}= lnC|x|[/tex].