1. Differential Equations Restricting the Domain of the Solution

2. Sketch the solution of the differential equation on the given slope field:  Note that while there are slopes to the left of the y-axis, our sketch can’t ever get that far because of the vertical slope at x = 0

3. Because the domain can only be one interval that contains the initial value. Find the solution of the differential equation: …and the domain is… Plug in initial value (1, 1)

4. Because the domain can only be one interval that contains the initial value. Sketch the solution of the differential equation on the given slope field:  Note that while there are slopes to the left of the y-axis, our sketch can’t ever get that far because of the vertical slope at x = 0

5. Sketch the solution of the differential equation on the given slope field:  Note that because we are dealing with positive slopes only, the sketch must stop just before the x-axis

6. One way to see why this is the domain would be to graph our solution Find the solution of the differential equation: …and the domain is… Plug in initial value (1, 4)

7. One way to see why this is the domain would be to graph our solution Sketch the solution of the differential equation on the given slope field: Note that to the left of –7, the solution graph crosses the slopes instead of following them.  And the derivative is undefined for y = 0 so we must exclude x = –7

8. Sketch the solution of the differential equation on the given slope field: Note that to the left of –7, the solution graph crosses the slopes instead of following them.  …and we would get negative slopes for x < –7 which contradicts the slope field Because derivative is undefined for y = 0, we must exclude x = –7

9. The domain to any solution to a differential equation is the largest open interval containing the initial value.  It also needs to satisfy any restrictions on the derivative as well When sketching the solution, the graph should never go against the slope field. Like a continuous function, you should never have to pick your pencil up off the paper when sketching the solution