A particle of mass m is moving in the xy -plane such that its...

As force is given by, $\overset{\to}{F} = m\dfrac{d\overset{\to}{v}}{dt}$.

Given: $\overset{\to}{v} = \alpha\left(y\overset{∧}{x} + 2x\overset{∧}{y}\right)$, where velocity along x-axis is ${v}_{x} = \alphay$ and velocity along y-axis is ${v}_{y} = 2x\alpha$.

Therefore,

$\dfrac{d\overset{\to}{v}}{dt} = \alpha\left(\dfrac{dy}{dt}\overset{∧}{x} + 2\dfrac{dx}{dt}\overset{∧}{y}\right)$

$= \alpha\left({v}_{y}\overset{∧}{x} + 2{v}_{x}\overset{∧}{y}\right)$

$= \alpha\left[2x\alpha\overset{∧}{x} + 2\alphay\overset{∧}{y}\right]$

$= 2{\alpha}^{2}\left[x\overset{∧}{x} + y\overset{∧}{y}\right]$

Therefore, required value of $\overset{\to}{F} = 2m{\alpha}^{2}\left[x\overset{∧}{x} + y\overset{∧}{y}\right]$.