Toronto Math Forum
APM3462022S => APM346Lectures & Home Assignments => Chapter 2 => Topic started by: Xiangmin.Z on January 17, 2022, 04:40:37 PM

Hello, I have a question about example 4 from W2 L1:
We are given :$u_{t}+xu_{x}=xt $
after calculation we get:
$x=Ce^t$
$du=xt dt=Cte^t dt$, so $u=C(t1)e^t+D=x(t1)+D$,
but how did we get $D=\phi({xe^{t}} )$? we know it is a constant, but why is D depended on $xe^{t}$?
Also, why would the initial condition $u_{t=0} =0 $ implies that $\phi({x}) =x$ ?
Thanks.

I checked the calculation and I think the answer for x is correct, and does anyone know why D is depended on $xe^{t}$?

Now it is correct $x=Ce^{t}$ and then $C=?$

$C=xe^{t}$, so D is a function of $\phi$, therefore $D=\phi({xe^{t}} )$ ? Yes, but "$D$ is a function of it"