anonymous
  • anonymous
differential equation help find the particular solution of the differential equation (dy/dx)=500-y that satisfies the initial condition y(0)=7
Mathematics
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
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...
anonymous
  • anonymous
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: \[y=500 + Ce ^{-x}\]
anonymous
  • anonymous
for a particular solution y(0), just plug in the 0 and find value of C for it

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anonymous
  • anonymous
got it?
anonymous
  • anonymous
c=-507
anonymous
  • anonymous
So particular solution would be:\[y _{p}= 500 -507e ^{-x}\]
anonymous
  • anonymous
@helpmenowplease
anonymous
  • anonymous
yea i got it thanks @myko
anonymous
  • anonymous
@myko wouldn't the C value be -493 not -507
anonymous
  • anonymous
ups, ya sry

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