Singular solution for a differential equation?

[shackdaddy836]

Hey, this is my first time on the forums so sorry if I do something wrong...
1.
Let c'(t) = f(c) = $\frac{(kr + P - c(t)r)}{V}$. Determine the singular solution of Vc'(t)


2.
k, P, r, and V are all constant


I'm not quite sure if this is the correct way to find the singular solution but I used this equation:
Integrating Factor Method:
To solve a linear DE
y' + x(t)y = f(t)

where x and f are continuous on the domain I:
u(t) = e$^\int{x(t) dt}$

u(t)[y' + x(t)y] = f(t)u(t)

$\frac{d}{dt}$[u(t)y(t)] = f(t)u(t)

u(t)y(t) = $\int{f(t)u(t )dt}$ + j

where j is a constant

y = ($\frac{\int{f(t)u(t) dt} + j)}{u(t)}$


3.
f'(c) = -$\frac{c'(t)r}{V}$

f'(c) = -$\frac{f(c)r}{V}$

f'(c) + $\frac{f(c)r}{V}$ = 0

let x = $\frac{r}{V}$

u = e$^{\int{xdt}}$

u = e$^{\frac{d}{dt}}$

$\frac{d}{dt}$[e$^{\frac{rt}{V}}$f(c)] = 0

e$^{\frac{rt}{V}}$f(c) = j

f(c) = $\frac{j}{e^\frac{rt}{V}}$

There is no singular solution


Now that can't be right can it? I'm completely confused and now I'm second guessing myself everywhere. Any help?

 

[mighty2000]

Hi there, welcome to the forums! Don't worry about doing something wrong, we're all here to learn and help each other out.

First, let's clarify what the singular solution is. In this case, it refers to a solution of the differential equation that cannot be obtained using the general solution. It is a special solution that arises from setting a constant value for the arbitrary constant in the general solution.

Now, let's look at the given equation:
c'(t) = f(c) = \frac{(kr + P - c(t)r)}{V}

To find the singular solution, we need to set a constant value for c(t) in the general solution. So let's assume that c(t) = k, where k is a constant. This means that c'(t) = 0.

Substituting this into the given equation, we get:
0 = f(k) = \frac{(kr + P - kr)}{V} = \frac{P}{V}

So, the singular solution is c(t) = k, where k is a constant and P and V are also constants. This means that the concentration, c(t), remains constant over time.

I hope this helps clarify the concept of singular solution. Let me know if you have any other questions. Good luck with your studies!