# Special First-Order Differential Equation with Two Constraints

Let

• $D =[0,\overline{c}] \subset \mathbb{R}_+$
• $\pi(c):D \to \mathbb{R}_+$ be continuously differentiable, bounded and strictly decreasing.
• $V(c):D \to \mathbb{R}$ be continuously differentiable, bounded and strictly increasing.
• $F:D \to [0,1]$ be a continuous cumulative distribution function and $f$ the corresponding density.
• $\tilde{c} \in D$ be the unique solution to the equation $V(c)=\pi(c)$.

I have the following first-order differential equation:

$V(c)-\left(\frac{1}{1-F(\tilde{c})}+\frac{f(c)}{2}\right)V'(c)=\frac{1}{2}\frac{F(\tilde{c})}{1-F(\tilde{c})}(f(c)\pi'(c)+\pi(c)).$

Furthermore, I know that $V(0) = \frac{\pi(0)}{2} - k$ where $k > 0$ is some exogenous parameter. Is the information I have sufficient to determine $V(c)$ and/or $\tilde{c}$?

If not: Is it at least possible to solve this for the special case $D=[0,1], f(c) = 1$ and $\pi(c) = 1-c$?

Any nudges in the right direction are welcome, e.g. hints if the problem needs additional structure to be solvable.