Let f be a differentiable function such that f^2(x+2) = 2f(x)-3(x+2)^2 = 10_0^x(t+2)f(t),d…

Question

Let $f$ be a differentiable function such that $f^2(x+2) = 2f(x)-3(x+2)^2 = 10\int_0^x(t+2)f(t)\,dt$, $x\ge0$. Then $f(2)$ is equal to \_\_\_\_\_\_.

Answer: 19

Accepted Answer

Differentiating and simplifying gives
$$(x+2)\frac{dy}{dx}-3y=3.$$

Solving: $\ln(y+1)=3\ln(x+2)+C \Rightarrow y+1=C(x+2)^3$.

Using $f(0)=\dfrac32$ as an intermediate condition, one obtains $f(2)=19$.

Solutions · 1