设f(x)在[0,+∞)上非负连续,且f(x)∫0xf(x-t)dt=2x3,则f(x)=________.

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问题 设f(x)在[0,+∞)上非负连续,且f(x)∫0xf(x-t)dt=2x3,则f(x)=________.

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答案2x

解析0xf(x-t)dt→∫0xf(u)(-du)=∫0xf(u)du,令F(x)=∫0xf(u)du,由f(x)∫0xf(x-t)dt=2x3,得f(x)∫0xf(u)du=2x3,即d/dx[1/2F2(x)]=2x3,则F2(x)=x4+C0.因为F(0)=0,所以C0=0,又由F(x)≥0,得F(x)=x2,故f(x)=2x.
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