The graph of the function f is shown in the first set of axe

The graph of the function f is shown in the first set of axes below. On the second, set of axes, sketch the graph of the solution of the initial value problem {dy/dx=f (x) y(0)=1 your graph should captivate the monotonicity and concavity of the solution.

Solution

From x=0 to x=5

The graph of y\' = x

dy/dx = x

Seperating the variables and integrating both sides we get

y = x^2/2 + C

Using the initial conditions we get y=x^2/2, from x=0 to x=5

From x=5 to x = 10

y\' = 10 - x

y = 10x - x^2/2 + C

The function must be continuous at x=5, hence we get

25/2 = 50 - 25/2 + C

C = -25

Hence the final solution to the differential equation will be

y = x^2/2 from x=0 to x=5

y = 10x - x^2/2 - 25 from x=5 to x=10