Differential Equation with Integration

Question

Function $f(x)$ is differentiable. Given further that

\begin{eqnarray*} f’(x)+xf’(x-1) &=& 4 \\ \int_0^1 f(xt) dt + \int_0^x f(t-1)dt &=& x^3 +x^2 +2x \end{eqnarray*}

Determine the function $f(x)$.

$\textbf{Attempt}$ I know $\frac{d}{dx} \int_0^x f(t-1)dt = f(x-1)$, but I am confused about what’s $\frac{d}{dx} \int_0^1 f(xt) dt $ . Maybe I can do $ - \frac{d}{dx} \int_1^x f(xt) dt - \frac{d}{dx} \int_0^x f(xt) dt $ ?

Answer 1

You can set $xt=u$.

Setting $xt=u$ gives $x=\frac{du}{dt}$, so $$\int_0^1 f(xt) dt =\int_{0}^{x}f(u)\cdot\frac{du}{x}$$ So, differentiating the both sides of $$\frac 1x\int_{0}^{x}f(u)du+ \int_0^x f(t-1)dt =x^3 +x^2 +2x$$ gives $$-\frac{1}{x^2}\int_{0}^{x}f(u)du+\frac{f(x)}{x}+f(x-1)=3x^2+2x+2,$$ i.e. $$\int_{0}^{x}f(u)du=xf(x)+x^2f(x-1)-3x^4-2x^3-2x^2$$ Now, differentiate the both sides and use $f'(x)+xf'(x-1)=4$.

Answer 2

For the proposer's work:

Using the Leibniz differential rule one has: $$\frac{d}{dx} \, \int_{a}^{b} f(x, t) \, dt = f(x, b) \frac{d b}{dx} - f(x,a) \frac{d a}{dx} + \int_{a}^{b} \frac{d f}{dx} \, dt,$$ where $a = a(x)$, and $b = b(x)$. In this example $$\frac{d}{dx} \, \int_{0}^{1} f(x t) \, dt = \int_{0}^{1} t \, f'(x t) \, dt.$$

Since $$\int_0^1 f(xt) dt + \int_0^x f(t-1)dt = x^3 +x^2 +2x$$ then differentiation would lead to $$\int_{0}^{1} t \, f'(x t) \, dt + f(x-1) = 3 x^2 + 2 x + 2.$$ Seemingly this leads to just as much "confusion".

Alternatively one can take the path of the following.

Using $$\int_{0}^{1} f(x t) \, dt = \text{ with } u = xt \text{ then } \int_{0}^{x} \frac{f(u) \, du}{x}$$ and \begin{align} \int_{0}^{1} f(x t) \, dt + \int_{0}^{x} f(t-1) \, dt &= g(x) = x^3 + x^2 + 2 x \\ \int_{0}^{x} ( f(t) + x \, f(t-1) ) \, dt &= x \, g \\ \frac{d}{dx} \int_{0}^{x} ( f(t) + x \, f(t-1) ) \, dt &= \frac{d}{dx}(x \, g) \\ f(x) + x \, f(x-1) + \int_{0}^{x} f(t-1) \, dt &= \frac{d(x g)}{dx} \end{align} Differentiation again leads to \begin{align} f'(x) + x \, f'(x-1) + f(x-1) + f(x-1) &= \frac{d^2(x g)}{dx^2} \\ 4 + 2 \, f(x-1) &= 2(6 x^2 + 3x + 2) \\ f(x-1) &= 6 x^2 + 3 x = 3x \, (2x +1) \\ f(x) &= 6 x^2 + 15 x + 9. \end{align}