Let f be a differentiable function such that x^2f(x)-x=4_0^x tf(t),dt, f(1)=23.

Question

Let $f$ be a differentiable function such that $x^2f(x)-x=4\int_0^x tf(t)\,dt$, $f(1)=\dfrac23$. Then $18f(3)$ is:

Accepted Answer

Differentiate: $x^2f'(x)+2xf(x)-1=4xf(x) \implies \dfrac{dy}{dx}-\dfrac2xy = \dfrac{1}{x^2}$ (with $y=f(x)$).

I.F. $=1/x^2$.
$$\frac{y}{x^2} = -\frac{1}{3x^3}+C.$$

Using $f(1)=2/3$: $C=1$. So $y = -\dfrac{1}{3x}+x^2$.

$$f(3) = -\frac19+9 = \frac{80}{9} \implies 18f(3)=160.$$

Solutions · 1