Question

Find the value of kk for which the constant function x(t)=kx(t)=k is a solution of the differential equation 3t3dxdt+5x−3=03t3dxdt+5x−3=0.

Answer 1

If $x(t)=k$ is constant, then $\dfrac{\mathrm dx}{\mathrm dt}=0$, so you have

$5k-3=0\implies k=\dfrac35$