The diagram shows the direction field of a differential equation - HSC - SSCE Mathematics Extension 1 - Question 3 - 2023 - Paper 1

To determine where the solution crosses the y-axis, we first need to consider that the y-axis is defined by the equation $x=0$. Thus, we will follow the direction field from the point (−2,1) to where $x=0$.

From the direction field shown in the diagram, we analyze the vector field at various points along the trajectory as we move horizontally from $x=-2$ toward $x=0$. We look for points where the direction changes and evaluate the slopes given by the arrows in the field.

Following the y-values as we approach the y-axis, the trajectory that starts at (−2,1) will lead us to approximately the point where $y$ is around $1.56$. This suggests that the solution should cross the y-axis at this value.

Therefore, the answer is:

C. y = 1.56.