Question

Solve the differential equation: $y-x\frac{dy}{dx}=0$

Answer 1

To solve the differential equation: $y-x\frac{dy}{dx}=0$,first separate the variables than integrate both sides

$y-x\frac{dy}{dx}=0 \\ \\ y = x\frac{dy}{dx} \\ \\ \int\frac{1}{x} dx = \int\frac{1}{y} dy \\ \\ integrate \: both \: side \\ \\ log \: x = log \: y + c \\ \\ we \: can \: write \: integration \: constant \: as \: log \: c \\ \\ log \: x = log \: y + log \: c \\ \\ log \: x = log \: cy \\ \\ remove \: log \: from \: both \: sides \\ \\ x = cy \\ \\ or \\ \\ y = \frac{x}{c} \\ is \: the \: answer$