Question

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

Answer 1

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.

Answer 2

\(y''=5y+6+\dots\)
Or whatever.

Answer 3

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.