Question

Solve the differential equation: $e^{{dy}/{dx}} = x$

Answer 1

To solve the differential equation:
$e^{{dy}/{dx}} = x$

first we have to remove exponential,by taking log both side

$log \: (e^{{dy}/{dx}}) = log \: x \\ \\ \frac{dy}{dx} = log \: x$

2) Now separate the variables

${dy} = log \: x \: dx$

3) Integrate both sides
$\int\:1.dy= \int\:log\:x\:dx \\ \\y =log\:x\int\:1.dx-\int[\frac{d\:log\:x}{dx}\int1.dx]dx + C\\\\y = (log\:x)x-\int[\frac{1}{x}(x)]dx + C\\\\y = (log\:x)x-x+ C\\\\ y = x[(log\:x)-1]+ C$
is the answer.