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^ (1)/(2)` which is
`(x^(3)/(2))/( (3)/(2) )` or
`(2x^(3)/(2))/(3)`
e^y=
`(20x^(3)/(2))/(3)` +C
take the ln of both sides

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

Answer 2

see photo for solution


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