Differential equation boundary conditions

[kasse]

$$\frac{dN}{dt}=-k_sN^2$$

Attempt:

$$\frac{1}{N^2}dN = -k_s dt$$

Integrate:

$$-\frac{1}{N} + C = -k_s t$$

In the solution manual, C is written $$\frac{1}{N_0}$$

Why?

 

[dx]

If you integrate from 0 to t on the right, you have to integrate from N₀ to N_t on the left, where N₀ is the value of N at t = 0.

 

[Cyosis]

Alternatively use the boundary conditions in this case $N(0)=N_0$ and then solve for C.