Question

differential equation help

find the particular solution of the differential equation (dy/dx)=500-y that satisfies the initial condition y(0)=7

Answer 1

what my first thought is that you would integrate with respect to y and i got this\[x=-\ln \left| 500-y \right|+C\] and i solved that, but i got 6.2005...

Answer 2

this is linear edo: so dy/dx +y=500
Integration factor:
\[e ^{\int\limits dx} =e ^{x}\]
multiply bouth sides of edo by this factor:
\[e ^{x}dy +ye ^{x}dx=500e ^{x}dx\]
left side is actualy a:\[d/dx[e ^{x}y]\]
so:
\[d/dx[e ^{x}y]=500e ^{x}dx\]
integrating:
\[e ^{x}y=500e ^{x}+C\]
rearanging:
\[y=500 + Ce ^{-x}\]

Answer 3

for a particular solution y(0), just plug in the 0 and find value of C for it

Answer 4

got it?

Answer 5

c=-507

Answer 6

So particular solution would be:\[y _{p}= 500 -507e ^{-x}\]

Answer 7

@helpmenowplease

Answer 8

yea i got it thanks @myko

Answer 9

@myko wouldn't the C value be -493 not -507

Answer 10

ups, ya sry