A bucket has total volume 2 L and 20 cm height which is separated into two equal volume side right and left by a partition in the middle.
A tap water has flow rate 100 mL per second (assume that water flow to left side instantly).
There is a hole at the bottom of the partition through which the water flows to the other side with flow rate given by $10\times$(Difference in height of water level in cm) mL per second.
If I open a tap water at t=0 when the bucket is empty, how long does it take for the left side to be filled, and in that time how much has the water level in the right side risen to?
My Attempt:
I let $A$ to be the height of the water in the left side and $B$ to be the height of the water in the right side. So:
$$50*A'(t)=100+10*(B(t)-A(t))$$ $$50*B'(t)=10*(A(t)-B(t))$$ $$\Downarrow$$ $$A'(t)=2+\frac{B(t)-A(t)}{5}$$ $$B'(t)=\frac{A(t)-B(t)}{5}$$
To find a general solution, let-
$$A(t)=a*e^{bx}$$ and $$B(t)=e^{bx}$$
Then, I get:
$$5*ab=1-a$$ $$5*b=a-1$$ $$\Downarrow$$ $$a=-1$$ $$b=-2/5$$
At this point, I can find homogeneous solution but don't know how to find a particular solution. How do I proceed?