• mew55
i need help. a differential equation problem. Find the general solution of the first order linear differential equation and use it to determine how solutions behave as t>infinity
  • Stacey Warren - Expert
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  • katieb
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  • theEric
Hi! I'm just staring to learn about ordinary differential equations, so I probably can't help. But you'd need to have your equation so we can see how it behaves. If you need help putting it into the fancy \(m+a+t+h\) writing, I can \(\dfrac{h\ e\ \quad\ }{\ \ \quad l\ p}\) because I am used to \(\large_i^{\quad t}\).... I hope you see what I mean about the fancy math writing.
  • theEric
\(y''=5y+6+\dots\) Or whatever.
  • ybarrap
The simplest type of 1st order linear diff equation is something like $$ \Large y'(t)+by(t)=c\\ $$ With solution: $$ \Large y=e^{-a(t)}\left (\int c e^{a(t)} dt+\kappa\right )\\ $$ where $$ \Large a(t)=\int b~ dt=bt\\ $$ So, $$ \Large y=e^{-bt}\left ({e^{bt}\over b}+\kappa\right )\\ \Large ~~~={1\over b}+e^{-bt}\kappa $$ So, as \(\Large t\to\infty,~y(t)\to\dfrac 1 b\) if \(\Large b>0\). Otherwise, it diverges.

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