# Analytical solution to this second order nonlinear ordinary differential equation

I'm trying to find an analytical solution to the following equation:

$$a_1 x''(t)+a_2x'(t)+a_3x(t)^\alpha+a_4t+a_5=0$$

where $$0<\alpha<1$$. Any suggestions would be greatly appreciated.

• It's unlikely to have closed-form general solutions. – Robert Israel Feb 6 at 18:51

I can give you some solutions in the case $$\alpha = 1/2$$. Namely, in this case $$x(t) = (b_1 t + b_0)^2$$ is a solution when $$b_1$$ is a root of the quadratic $$2 a_2 b_1^2 + a_3 b_1 + a_4 = 0$$ and $$b_0 = -{\frac {2\,{a_1}\,{{ b_1}}^{2}+{ a_5}}{2\,{ a_2}\,{ b_1}+{ a_3}}}$$ and $$b_1 t + b_0 > 0$$.
• Thank you. Do you think of there are other solutions in the case $\alpha=\frac{1}{2}$? – user64735 Feb 6 at 19:41
• Thanks. Would there be some value of $0<\alpha<1$ for which closed-formed general solutions can be obtained? – user64735 Feb 6 at 20:47
• Of course you can't do that in general. What if $f = x$? What if $f = x + 1$? – Robert Israel Feb 7 at 1:48