differential equations

The German word you would use in this context is  "Bitte" or "Bitte sehr"

"ihr willkommen" is equivelent in the context of "welcome" to my home.

is there a name of these kind of questions ? 

Sure,

they are called non-autonomous linear first order differential equations.

okay , the name sounds a bit scary but ill try to find out ! all the best for ur search !

Here's how it's done for your simple example.  It isn't always possible to get nice results for situations where a(t) and b(t) are very complicated.  For those, use a numerical method!

I need to correct one item above.  In general the integrating factor is $$e^{\int{-a(t)}dt}$$  not $$e^{-a(t)t}$$.

Just noticed Bertie has said this below.

see my " all the best " works ! alan came so fast for ur helt after my " all the best " isnt it ! 

The usual approach is to use what is known as an Integrating Factor.

For your example, start by moving a(t)y(t) to the lhs and then multiply throughout by

$$\displaystyle \exp\{\int-a(t)dt\}$$

The lhs will then be a perfect derivative,

$$\displaystyle \frac{d}{dt}\left[\exp\{\int-a(t)dt\}.y(t)\right]$$

and, (hopefully), you can then integrate both sides, having already (hopefully), integrated a(t).

Thank you Alan 

So I gave it a try (not really involving a(t) yet)

and this is what I have

$$\begin{array}{rll} \frac{dy}{dt} = y-t^2\\
\frac{dy}{dt} - y = -t^2\\
e^{-t}\frac{dy}{dt}-e^{-t}y = -e^{-t}t^2\\
\frac{d}{dt}(e^{-t}y) = -e^{-t}t^2\\
\end{array}$$

$$\begin{array}{rll} e^{-t}y = e^{-t}(t^2+2t+2)+c\\
\mbox{Since }y(0)=y_0\\
e^0y_0 = e^0(0^2-2*0+2)+c\\
c = y_0-2\\
y(t) = t^2+2t+2+y_0-2\\
y(t) = t^2+2t+y_0
\end{array}$$

Does that make sense? 

edit: (I also did it for a(t) = t, but I just found my mistake)

Almost, the c = (y₀ - 2) should be multiplied by e^t.

Why is that?

e^-ty = e^-t(t² + 2t + 2) + c, where c = y₀ - 2.

Multiply throughout by e^t.

Bertie is right.

Also, in the line below "Since y(0) = y₀" you have the term -2*0.  This should be +2*0.  However, since the result is zero anyway, this doesn't have any impact on the result!

Yes, that makes sense.

Thank you!

reinout u didnt say anything about my " all the best " ? i think u forgot it ! 

Haha, yes that was very helpful!

Dankjewel (That's Dutch for thank you)

ihr willkommen (thats german for ur welcome )

Some fine german lessons here!

i have german lessons in my class too reinout ! but uh , i have to study german in here too !