Challenge Level

Find the function $f(x)$ which solves the equation $$\int_0^x f(t)\,dt = 3f(x)+k\,,$$ where $k$ is a constant.

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In the same way that a differential equation is formed from differentials, an integral equation is formed from integrals. Problems in mathematics might naturally be specified in terms of integrals, others in terms of differentials. Differentials are mainly used when the problem involves only local changes, such as the force at a point, whereas integrals are used where the problem involves a global property of a system, such as the total energy.

In the same way that a differential equation is formed from differentials, an integral equation is formed from integrals. Problems in mathematics might naturally be specified in terms of integrals, others in terms of differentials. Differentials are mainly used when the problem involves only local changes, such as the force at a point, whereas integrals are used where the problem involves a global property of a system, such as the total energy.