What is the particular solution to the differential equation dy/dx=(x+5)/y with the initial condition y(-4)=-6

This equation is separable:

dy/dx = (x + 5) / y   →   y dy = (x + 5) dx

Integrate both sides to get

1/2 y² = 1/2 (x + 5)² + C … … … (if you substitute u = x + 5)

or

1/2 y² = 1/2 x² + 5x + C … … … (if you integrate term-by-term)

Given that y (-4) = -6, meaning x = -4 and y = -6, plug this in to solve for C. Note that whichever method you choose determines the particular value of C.

1/2 (-6)² = 1/2 (-4 + 5)² + C

18 = 1/2 + C

C = 35/2

→   1/2 y² = 1/2 (x + 5)² + 35/2

or

1/2 (-6)² = 1/2 (-4)² + 5(-4) + C

18 = 8 - 20 + C

C = 30

→   1/2 y² = 1/2 x² + 5x + 30

Both implicit solutions are correct and equivalent; the second one is just the expanded form of the first.

If you want an explicit solution, you can solve for y. But keep in mind that the initial condition tells you y < 0, so you should take the negative square root.

1/2 y² = 1/2 (x + 5)² + 35/2

y² = (x + 5)² + 35

y = -√((x + 5)² + 35)

or

y = -√(x² + 10x + 60)