Solve differential equation $y' = |1.1 - y| + 1$

Question

How can the following differential equation be solved analytically?

\begin{equation*} y' = |1.1 - y| + 1, \\ y(0) = 1. \end{equation*}

I guess one must rewrite the differential equation piecewise and solve each piece independently. But how and how do I continue?

Thanks in advance for your assistance.

Answer 1

Solving the two differentail equations in Paul's post yields:

\begin{equation*}y(x) = 2.1 - 1.1e^{-x},|1.1−y| > 0 \end{equation*} and

\begin{equation*}y(x) = e^{x} + 0.1, |1.1−y| < 0 .\end{equation*}

Answer 2

Break the time into 2 periods.

(1) Initially, the argument of the absolute value is positive. In this case, you get

$$y'=2.1-y$$ which can be solved by standard methods.

(2) When you get a solution here, find out at what time $y=1.1$ (call that time $t_0$). At $t_0$ there will be a change in behavior. $y'$ will still be positive, so immediately thereafter, the argument of the absolute value will be negative, so $$y'=y-0.1$$ with initial condition $y(t_0)=1.1$. This in turn can be solved normally. And since $y'>0$ for all $t>t_0$, you don't have to worry about crossing $y = 1.1$ again. So you only have 2 periods to work with, and this gives you the solution.