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Problem Given

f(x) = e^x + \int\limits_{0}^{x} (t-x)f(t)\;dt\quad\quad\quad(\star)

where f(x) is a continuous function, find f(x).

Solution

From (\star), we see that

f(0) = 1;

f(x) = e^x + \int\limits_{0}^{x} t\cdot f(t) - x\cdot f(t) \;dt = e^x + \int\limits_{0}^{x} t\cdot f(t)\;dt-x\cdot \int\limits_{0}^{x}f(t)\;dt.

And so,

\frac{df(x)}{dx}=\frac{de^x}{dx} + \frac{d}{dx}\int\limits_{0}^{x}tf(t)\;dt - \frac{d}{dx}\left(x\cdot \int\limits_{0}^{x}f(t)\;dt\right)

=e^x+\frac{d}{dx}\int\limits_{0}^{x}tf(t)\;dt-\left(\int\limits_{0}^{x}f(t)\;dt + x\frac{d}{dx}\int\limits_{0}^{x}f(t)\;dt\right)

\overset{\textbf{FTC}}{=}e^x + xf(x) -\int\limits_{0}^{x} f(t)\;dt - xf(x)

= e^x - \int\limits_{0}^{x}f(t)\;dt

That is,

\frac{d}{dx}f(x)= e^x - \int\limits_{0}^{x}f(t)\;dt\implies f'(0) = 1.

It follows that

\frac{d^2f(x)}{dx^2}=\frac{d}{dx}\left(e^x-\int\limits_{0}^{x}f(t)\;dt\right)=e^x-\frac{d}{dx}\left(\int\limits_{0}^{x}f(t)\;dt\right)\overset{\textbf{FTC}}{=}e^x-f(x).

Solving

\begin{cases} f''(x)=e^x-f(x) \\f(0)=1\\f'(0)=1 \end{cases}\quad\quad\quad(\star\star)

gives

f(x) = \frac{1}{2}(\sin(x)+\cos(x)+e^x).

Fig. 1

Notice the derivation of (\star\star) can be simplified if Leibniz’s Rule (LR-1, see “A Missing Piece from Popular Textbooks”) is applied:

\frac{df(x)}{dx} = e^x + \underline{\frac{d}{dx}\int\limits_{0}^{x}(t-x)f(t)\;dt}

\overset{\textbf{LR-1}}{=} e^x + \underline{\int\limits_{0}^{x}\frac{\partial}{\partial x}(t-x)f(t)\;dt}

=e^x+\int\limits_{0}^{x}-1\cdot f(t)\;dt

= e^x-\int\limits_{0}^{x}f(t)\;dt

\implies \frac{d^2f(x)}{dx}=e^x-\frac{d}{dx}\int\limits_{0}^{x}f(t)\;dt\overset{\textbf{FTC}}{=}e^x-f(x).

Fig.2 shows that Omega CAS explorer‘s Maxima engine is both FTC and LR-1 aware:

Fig. 2


Exercise-1 Given:

f(x) = \int\limits_{0}^{x}t\cdot f(x-t)\;dt+\sin(x)

where f(x) is a continuous function, find f(x).

hint: Let u=x-t, t = x-u; t=0\implies u=x; t=x\implies u=0; \frac{du}{dt}=-1;

f(x) = \int\limits_{x}^{0}(x-u)\cdot f(u)\cdot (-1)\;du + \sin(x)=\int\limits_{0}^{x}(x-u)f(u)\;du+\sin(x).

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