MHB Finding $f$ When $6\int_{1}^{x} f(t)\, dt+5=3x \, f(x)-x^3$

[sbhatnagar]

Let $f:[1,\infty)\to [2,\infty)$ be a differentiable function such that $f(1)=2$. If

$$ 6\int_{1}^{x} f(t)\, dt+5=3x \, f(x)-x^3$$

for all $x \geq 1$, then:

1) Find the value of $f(2)$.

2) Find $\mathcal{L} \{ f(t)\}$.

 

[Opalg]

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 $\mathcal{L} \{ f(t)\}$.

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]

Sorry, I made a typo. The equation is

$$6\int_{1}^{x}f(t)dt+5=3xf(x)-x^3$$

I am really sorry about this.(Sadface)

 

[chisigma]

... the equation is...

$$6\int_{1}^{x}f(t)dt+5=3xf(x)-x^3$$

...

Deriving both terms You arrive to the linear first term ODE...

$\displaystyle \frac{d}{d x}\ f(x)= \frac{f(x)}{x} + x$ (1)

… with initial condition' $f(1)=0$ and the solving procedure is 'standard'...

Kind regards

$\chi$ $\sigma$