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g(x) phi(x) = integral of (h(t) - k phi'(t)): differentiate to get a linear equation in phi

This type of question, asked 2 times: 2021 Paper 22, 31 Jan 2023. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 22

    88
    If xϕ(x)=∫5x(3t2−2ϕ′(t))dt,x>−2x\phi(x) =\int_{5}^{x} \left( 3t^{2}- 2\phi^{'}(t) \right)dt,x> - 2, and ϕ(0)=4\phi(0) = 4, then ϕ(2)\phi(2) is

  2. Question 2

    31 Jan 2023

    169
    If ϕ(x)=1x∫π4x(42sin⁡t−3ϕ′(t))dt,x>0\phi(x) =\frac{1}{\sqrt{x}}\int_{\frac{\pi}{4}}^{x} \left( 4\sqrt{2}\sin t - 3\phi^{'}(t) \right)dt,x > 0, then ϕ′(π4)\phi^{'}\left( \frac{\pi}{4} \right) is equal to :

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