Question

Solve the differential equation. (dy)/(dx) = 10 sqrt(x)/e^y

Answer 1

We will do seperation of variables

group y's with dy and x  with dx
integrate both sides, and add constant to one side
use initial condiiton (if applcable)
solve for y



so

dy/dx=10(√x)/(e^y)
times both sides by e^y times dx
e^y dx=10(√x) dx
integrate both sides
an antiderivitive of 10(√x) is 10 times the antiderivitive of $x^ \frac{1}{2}$ which is $\frac{x^\frac{3}{2}}{ \frac{3}{2} }$ or $\frac{2x^\frac{3}{2}}{3}$
e^y=$\frac{20x^\frac{3}{2}}{3}$ +C
take the ln of both sides
$y=ln(\frac{20x^\frac{3}{2}}{3}+C)$

Answer 2

See photo for solution


y = ln(20x^(3/2) / 3 + C)