I have the following system of differential equations:
$$ \begin{cases} x'(t)(1-y(t))=0.1\delta(t-1)\\ y'(t)(1-x(t))=0.1\delta(t-1) \end{cases} $$ where $\delta(t)$ is the Dirac delta function upholding $\delta(t)=0$ for $t\neq0$ and $$ \int_{a}^{b}\delta(t)f(t)dt=f(0), \quad if\quad a<0<b $$ for every integrable function f(t).
Furthermore I know that both x(t) and y(t) are mass functions (in the sense that they are non-decreasing functions upholding $x(0)=y(0)=0$, $x(\infty)=y(\infty)=1$, and $x(t)=y(t)=0$ for $t<0$).
How do I prove that there are no functions $x(t)$ and $y(t)$ that can solve the system of linear equations (or if possible find a solution)?.
For $t<1$ and $t>1$ the differential equations require that $x$ and $y$ are both constant. The initial conditions make $x(t)=y(t)=0$ for $t<1$ and $x(t)=y(t)=1$ for $t>1$. But then $x'(t)=y'(t)=\delta(t-1)$ and we can not get $0.1\delta(t-1)$. There's also a problem multiplying $\delta(t-1)$ with $H(t-1)$ which has a jump discontinuity at $t=1.$