 Solution

At $r=1/2$, write $F(u)=-u^2(1-u{)}^2/4$, so $F^{\prime }=f$. Multiplying the stationary equation $m{U}^{\prime \prime }+F^{\prime }(U)=0$ by $U^{\prime }$ gives $m(U^{\prime }{)}^2/2+F(U)=C$. The endpoint values and vanishing endpoint derivatives make $C=0$. For the decreasing connection,Separate variables and absorb the integration constant into a translation $x_0$. The logistic front of a bistable cubic equation isIt approaches $1$ on the left and $0$ on the right and satisfies the stationary differential equation directly.