Let f:[1,)[2,) be a differentiable function satisfying 10_1^x f(t),dt = 5xf(x)-x^5-9 for x…

Question

Let $f:[1,\infty)	o[2,\infty)$ be a differentiable function satisfying $10\displaystyle\int_1^x f(t)\,dt = 5xf(x)-x^5-9$ for $x\ge1$. Then $f(3)$ is:

Accepted Answer

Differentiating: $10f(x) = 5f(x)+5xf'(x)-5x^4 \implies 5f(x) = 5xf'(x)-5x^4 \implies f(x)=xf'(x)-x^4$.

With $y=f(x)$: $\dfrac{dy}{dx}-\dfrac{y}{x} = x^3$. Integrating factor $1/x$:

$$\frac{y}{x} = \int x^2\,dx = \frac{x^3}3+C \implies f(x) = \frac{x^4}3+Cx$$

From the original relation at $x=1$: $0 = 5f(1)-1-9 \implies f(1)=2$. So $2=\dfrac13+C \implies C=\dfrac53$.

$$f(x) = \frac{x^4}3+\frac53x,\qquad f(3) = 27+5 = 32$$

Solutions · 1