The maximum value of $f(x)$ $f(x)$ is twice differentiable function such that $f(x)+f''(x)=-x\big|sinx\big|f'(x)$ where $x\geq0.$Given that $f(0)=-3,f'(0)=4$ the maximum value of $f(x)$ is$?$
My Try:To say, I was never taught these second order DE's but this question appeared under the topic maxima-minima,however I am familiar with first order DE's.Any particular way to solve these type of questions(I mean second order DE's)$?$

$$ff'+f'f''=-x|\sin x|(f'(x))^2$$
$$d(f^2+(f')^2) \leq 0$$
can you continue after this
