Chapter 1.5, Problem 43P

Figure 1.5.7 shows a slope field and typical solution curves for the equation $y^{\prime }=x-y$. (a) Show that every solution curve approaches the straight line $y=x-1$ as $x\to +\infty$. (b) For each of the five values $y_1=3.998,3.999,4.000,4.001,$ and 4.002, determine the initial value $y_0$ (accurate to four decimal places) such that $y\left(5\right)=y_1$ for the solution satisfying the initial condition $y\left(-5\right)=y_0$ .