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Find a function and constant such that a given integral equation holds

by RoRi

Find a function f and a constant c such that

    \[ \int_0^x f(t) \, dt = \int_x^1 t^2 f(t) \, dt + \frac{x^{16}}{8} + \frac{x^{18}}{9} + c \qquad \text{for all } x \in \mathbb{R}. \]

Let P(x) = \int_0^x t^2 f(t) \, dt. Then, taking derivatives, and using the fundamental theorem of calculus we have

    \begin{align*} && f(x) &= (P(1) - P(x))' + 2x^{15} + 2x^{17} \\ \implies && f(x) &= -x^2 f(x) + 2x^{15} + 2x^{17} \\ \implies && f(x)(x^2+1) &= 2x^{15} (x^2+1) \\ \implies && f(x) &= 2x^{15}. \end{align*}

Then, to find the value of the constant, let x = 0 and evaluate

    \begin{align*} &&0 &= \int_0^1 t^2 f(t) \, dt + c \\ \implies && c &= -\int_0^1 t^2 f(t) \, dt \\ \implies && c &= -\int_0^1 2t^{17} \, dt \\ \implies && c &= - \frac{1}{9} t^{18} \bigr \rvert_0^1 \\ \implies && c &= -\frac{1}{9}. \end{align*}

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