Solve \(\dfrac{{dy}}{{dx}} = 2x + 4\) given that \(P(2,-1)\) lies on the curve

The adjacent graph is the direction field of a first order differential equation. The differential equation could be: 

Given the differential equation \(\dfrac{{dy}}{{dx}} - y + x = 0\) and given that the solution curve passes through the point \((2,-1)\), then the gradient of the solution curve is?

The particular solution to \(\dfrac{{dy}}{{dx}} = x{({x^2} - 4)^3}\), given that \(y=1\) and \(x=2\)

Find the general solution to \(y' = 2{\sin ^2}x\)

Find the particular solution to \(\dfrac{{dy}}{{dx}} = {\sin ^2}y\), given that \(x=0\) and \(y = \dfrac{\pi }{2}\)

Find the general solution to \(y' = 1 + y\)

Find the particular solution to \((1 - {x^2})\dfrac{{dy}}{{dx}} = xy\), given that \(x=0,y=1\)

Find the general solution to \(\sec x\dfrac{{dy}}{{dx}} - {y^2} = 0\)

For a body falling under gravity, the rate of change of velocity is given by \(\dfrac{{dv}}{{dt}} = - 0.02(v - 490)\). Find the velocity after 5 seconds, given that when \(t=0,v=0\). 

The rate of cooling of a metal object, initially at \(T = 125^\circ C\), is given by \(\dfrac{{dT}}{{dt}} = - 0.64(T - 30)\), where \(t\) is in hours. Find how long it would take for the object to cool to \(66^\circ C\)