The rate of which certain substance dissolves in water is proportional at the product of the amount undissolved and the difference between $c_1-c_2$ where $c_1$ is the concentration of the saturated solution and $c_2$ is the concentration of the actual solution! If saturated $50gm$ of water will dissolve $20gm$ of the substance! If $10gm$ of the substance is placed in the $50gm$ of water then half of the substance will dissolve within 90 minutes! How much will dissolve in $3$ hours!
I have tried to solve this but no idea how to tackle this!
My attempt is below
$$\frac{dx}{dt} \alpha (c_1-c_2)$$
I try to introduce a proportionality constant, $k$ to the differential equation
Can someone hint me? And provide the right way to think of this problem.
Call x the amount of undissolved substance. Then $\frac{dx}{dt}$ gives you the rate at which the substance dissolves. This rate is proportional with the product of the amount of undissolved substance $x$ and $(c_1 -c_2)$
Hence, $$\frac{dx}{dt} = kx(c_1-c_2)$$
for some number $k \in \mathbb{R}$
Equivalently: $$\frac{dx}{x} = k(c_1-c_2)dt \iff \ln|x| = k(c_1-c_2)t + C $$
$$\Rightarrow x(t) = Ce^{k(c_1-c_2)t}$$
Can you take it from here?