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)

yes because there is exactly one y value for every x value

for function there can't be more than one value for every x value

after going and makeing graphs i would go with a,

there is nothing that is matches the graph. if there was i would show you. but therefore my answer is a.

hope i

-lavira: )