Direction Fields — Question 1

Question 1 – Solution

Strategy. Evaluate $x-y$ at points, then hold its value fixed to find isoclines. Test a proposed curve by comparing its derivative with the assigned slope.

Step 1: Slopes and regions. The four slopes, in the listed order, are $$\boxed{-2,\quad-1,\quad 0,\quad 2.}$$ Thus the first two segments fall to the right, the third is horizontal, and the fourth rises. The isocline equation is $x-y=k$, or $y=x-k$. Hence $$k=-1:\ y=x+1,\qquad k=0:\ y=x,\qquad k=1:\ y=x-1.$$ Below $y=x$ the slopes are positive; above it they are negative.

Step 2: Isoclines are not automatically solutions. Each displayed isocline has geometric slope $1$. On $y=x+1$ and $y=x$, the field instead prescribes $-1$ and $0$, so these lines are not solutions. On $y=x-1$, the prescribed slope is $1$, so that particular isocline also happens to be a solution.

Step 3: Tangent versus solution. At $(0,1)$ the slope is $-1$, giving $\boxed{y=1-x}$ as the tangent line. The solution initially falls. Along this line, however, $x-y=2x-1$, which equals the line’s derivative $-1$ only at $x=0$. Thus the tangent line does not solve the equation on any open interval.