Step 1: Rewrite the given equation. Using Newton-Leibniz theorem, we start with:

$3f\left(x\right)=x{f}^{\prime }\left(x\right)+f\left(x\right)-x^2$.

Step 2: Simplify and rewrite as a differential equation. Rearranging terms:

$f^{\prime }\left(x\right)+\frac{1}{x}f\left(x\right)=x$.

This is a first-order linear differential equation.

Step 3: Determine the integrating factor (I.F.). The integrating factor is:

I.F. = $e^{\int \frac{1}{x}dx}=e^{logx}=x$.

Step 4: Solve the differential equation. Multiply through by the integrating factor x:

$x{f}^{\prime }\left(x\right)+f\left(x\right)=x^2$.

This simplifies to:

$\frac{d}{dx}\left(xf\right(x\left)\right)=x^2$.

Integrating both sides:

$xf\left(x\right)=\int x^{2}dx=\frac{x^3}{3}+c$,

where c is the constant of integration.

Divide through by x:

$f\left(x\right)=\frac{x^2}{3}+\frac{c}{x}$.

Step 5: Apply the initial condition f(1) = 1/3. Substitute x = 1 and f(1) = 1/3:

$f\left(1\right)=\frac{1^2}{3}+\frac{c}{1}=\frac{1}{3}$.

Simplify:

1/3 + c = 1/3 ⇒ c = 0.

Thus, the solution becomes:

$f\left(x\right)=\frac{x^2}{3}$.

Step 6: Evaluate f(e). Substitute x = 4e into f(x):

$f\left(e\right)=\frac{4e^2}{3}$.

Final Answer:

$f\left(e\right)=\frac{4e^2}{3}$.

Quick Tip
For first-order linear differential equations, always compute the integrating factor and multiply through before solving.