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Find functions f(x) continuous on the real line satisfying given conditions

by RoRi

Find functions f(x) continuous everywhere on \mathbb{R} such that

    \[ f(x) f'(x) = 5x, \qquad f(0) = 1. \]

Let y = f(x), then we have a separable first order equation and we can compute,

    \begin{align*} f(x) f'(x) = 5x && \implies && yy' &= 5x \\ && \implies && \int y \, dy &= \int 5x \, dx \\ && \implies && \frac{1}{2}y^2 &= \frac{5}{2} x^2 + C \\ && \implies && y &= \sqrt{5x^2 + C} \\ && \implies && f(x) &= \sqrt{5x^2 + C}. \end{align*}

Then, since we are given the condition f(0) = 1 we have

    \[ f(0) = \sqrt{C} = 1 \quad \implies \quad C = 1. \]

Therefore,

    \[ f(x) = \sqrt{5x^2 + 1}. \]

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