The discussion revolves around finding the function \( f \) defined on the interval \([1, \infty)\) that satisfies the equation \( 6\int_{1}^{x} f(t)\, dt + 5 = 3x \, f(x) - x^3 \) for all \( x \geq 1 \). Participants explore the implications of this equation, including specific evaluations and transformations related to the function.
There is no consensus on the correctness of the initial conditions or the derived ODE, as participants have conflicting interpretations of the problem and its requirements.
The discussion reveals a potential inconsistency regarding the initial condition \( f(1)=2 \) versus \( f(1)=0 \) as derived in Post 4. The implications of these conditions on the solution remain unresolved.
Something wrong here? If $x=1$, then the equation $\displaystyle 6\int_{1}^{x} f(t)\, dt=3x \, f(x)-x^3$ becomes $0=5.$sbhatnagar said:Let $f:[1,\infty)\to [2,\infty)$ be a differentiable function such that $f(1)=2$. If
$$ 6\int_{1}^{x} f(t)\, dt=3x \, f(x)-x^3$$
for all $x \geq 1$, then:
1) Find the value of $f(2)$.
2) Find $\mathcal{L} \{ f(t)\}$.
sbhatnagar said:... the equation is...
$$6\int_{1}^{x}f(t)dt+5=3xf(x)-x^3$$
...