Global maxima of $f(x)$

We are being asked to consider the solution of the second order differential equation $f''(x) + x|\sin x|f'(x) + f(x) \; = \; 0 \;, \qquad \qquad x \ge 0 \;,$ together with the initial conditions $f(0) = -3$ and $f'(0) = 4$ . This is a case of strangely damped SHM; energy considerations tell us that $f(x)$ must tend to $0$ as $x \to \infty$ . Define $A(x) \; = \; 2\int_0^x u|\sin u|\,du \;, \qquad \qquad x \ge 0 \;,$ noting that $A(x)$ is a continuously differentiable increasing function of $x \ge 0$ with $A(0) = 0$ and $A'(x) \,=\, 2x|\sin x| \ge 0$ for all $x \ge 0$ . Then $\frac{d}{dx}\big[f(x)^2 + f'(x)^2\big] \; = \; 2f'(x)(f''(x) + f(x)) \; = \; -A'(x)f'(x)^2 \; ,$ which means that $f(x)^2 + f'(x)^2$ is a decreasing function of $x \ge 0$ . Thus $f(x)^2 \; \le \; f(x)^2 + f'(x)^2 \; \le \; f(0)^2 + f'(0)^2 \; = \; 25 \;, \qquad x \ge 0 \;,$ and so $|f(x)| \le 5$ for all $x \ge 0$ . Moreover $\begin{array}{rcl} \displaystyle \tfrac{d}{dx}\Big[e^{\frac12A(x)} f'(x)\Big] & = & \displaystyle e^{\frac12A(x)}\big[f''(x) + \tfrac12A'(x)f'(x)\big] \; = \; -e^{\frac12A(x)}f(x) \\ \displaystyle e^{\frac12A(x)}f'(x) & = & \displaystyle 4 - \int_0^x e^{\frac12A(u)} f(u)\,du \end{array}$

If $\xi$ is a turning point of $f(x)$ , then we must have $\int_0^\xi e^{\frac12A(u)}f(u)\,du \; = \; 4 \;.$ and $f(x)^2 \; \le \; f(x)^2 + f'(x)^2 \; \le \; f(\xi)^2 + f'(\xi)^2 \; = \; f(\xi)^2 \;, \qquad x \ge \xi \;,$ so that $|f(x)| \le |f(\xi)|$ for all $x \ge \xi$ . There are thus two possibilities:

• If $f$ has no turning points, then $f'(x) >0$ for all $x \ge 0$ , and $f(x)$ tends monotonically to $0$ . In this case $f(x)$ would have no maximum value, but would have least upper bound $0$ .
• If $f$ has a turning point $\xi > 0$ , then the fact that $\int_0^\xi e^{\frac12A(u)}f(u)\,du \; = \; 4$ means that $f(x)$ must be positive somewhere in the interval $0 \le x \le \xi$ . Thus $f(x)$ must have a strictly positive maximum value for $0 \le x \le \xi$ and hence for all $x \ge 0$ . This maximum value of $f(x)$ will occur at the first turning point of $f$ , and would lie between $0$ and $5$ .

Numerical solution of the differential equation in the interval $0 \le x \le \pi$ shows that the second of these two options occurs; the function $f(x)$ achieves its maximum value of approximately $\boxed{2.7319696}$ at the point $x = 1.8246435$ .