General solution of a differential equation $x''+{a^2}x+b^2x^2=0$ How do you derive the general solution of this equation:$$x''+{a^2}x+b^2x^2=0$$where a and b are constants.
Please help me to derive solution thanks a lot.

Note that the derivative of $(x')^2$ (respectively, $x^2$, $x^3$) is $2x'x''$ (respectively, $2xx'$, $3x^2x'$). Multiplying by $6x'$, we get $$(3)(2x'x'')+(3a^2)(2xx')+(2b^2)(3x^2x')=0$$ and integrating $$3(x')^2+3a^2x^2+2b^2x^3=C$$ Then $x'=\sqrt{C-3a^2x^2-2b^2x^3}/3$, which is variables-separable. But the integral it leads to is not elementary; as Graham notes, it leads to elliptic functions. 
