can you solve $x'+x''=\sqrt {x}$

Question

And is there a way to solve $$ ax''+bx'=f(x)?$$ If there is, how is the method called?

Answer 1

There is a way to reduce such equations (in which the independent variable $t$ does not appear explicitly) to the integration of first order equations.

Let $p=x'$ and consider $p$ as a function of $x$. Then $$ x''=\frac{dp}{dt}=\frac{dp}{dx}\,\frac{dx}{dt}=p\,\frac{dp}{dx}. $$ The equation becomes $$ a\,p\,\frac{dp}{dx}+b\,p=f(x),\quad\text{or}\quad\frac{dp}{dx}=\frac{f(x)}{a}\,\frac1p-\frac{b}{a}. $$ Unfortunately this equation does not always have an easy solution. If you find $p=p(x)$, you still have to solve $$ \frac{dx}{dt}=p(x), $$ a first order equation whose solution is $$ \int\frac{dx}{p(x)}=t+C. $$ Again, in a lot of cases the integral does not admit a closed form solution in terms of elementary functions.