# Singular solution for a differential equation?

1. Sep 27, 2011

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?