Differential Equations: JEE Main 2026 Mathematics question

Let $f:\mathbf{R}\to\mathbf{R}$ be such that $f(xy)=f(x)f(y)$, for all $x,y\in\mathbf{R}$ and $f(0)\neq0$. Let $g:[1,\infty)\to\mathbf{R}$ be a differentiable function such that $x^2g(x)=\displaystyle\int_1^x\left(t^2f(t)-tg(t)\right)dt$. Then $g(2)$ is equal to :