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

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] Differentiate: $4(x+2)f+2(x+2)^2f'-6(x+2)=10(x+2)f$, so $(x+2)f'=3f+3$, i.e. $\frac{f'}{f+1}=\frac3{x+2}$ and $f+1=C(x+2)^3$. At $x=0$: $8f(0)-12=0\Rightarrow f(0)=\frac32$, so $\frac52=8C\Rightarrow C=\frac5{16}$. $f(2)=\frac5{16}\cdot64-1=19$. Answer: 19.

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

Explanation

Reviewed by Dr. Abubacker

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