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I was trying to calculate an equation of motion and there was a differential equation I couldn't solve: $$ \frac{d^2\theta}{dt^2}+A\cos(\theta)=0 $$ Does someone know how to solve it?

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    The solution to this equation can't be given in terms of elementary functions, it requires elliptic integrals. In the limit that $\theta$ is small however you can find such a solution. ALthough it looks similar to a pendulum equation to me – 2017-01-31
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    It can be solved approximately atleast for some intial conditions. For example if $\theta(0) \approx \frac{\pi}{2}$ then taking $\theta = \pi/2 + \delta\theta$ we get $\ddot{\delta\theta} - A\delta\theta \approx 0$ so if $A<0$ we have stable oscillations $\delta\theta \approx \sin(\sqrt{-A}t + \phi)$. – 2017-01-31

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First, I would substitute $\theta=\pi/2-y$ making the equation $$-\frac{d^2y}{dt^2}+A \sin(y)=0 .$$ Secondly I would scale time according to the rule $t=x/\sqrt{A}$ (here I will assume that $A>0$). This will result in the simpler equation $$\frac{d^2y}{dx^2}=\sin(y) $$ The general solution to this equation, is similar (note the minus sign) to the one that appears in a previous question: $$y(x)=2\text{am}\left(c_1x +c_2\bigg|-\frac{1}{c_1^2}\right) $$ where am is the Jacobi amplitude function, and $c_1,c_2$ are arbitrary constants. Undoing the substitutions, we obtain the solution of the original equation as $$\theta(t)=\frac{\pi}{2}-2\text{am}\left(c_1\sqrt{A} t +c_2\bigg|-\frac{1}{c_1^2}\right) .$$

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This is like duffing's equation ,for little value of $\theta$

$\frac{d^2\theta}{dt^2}+A\sin (\theta)=0$
to find solution :put $sin (\theta)=\theta-\dfrac{\theta^3}{3!} $

and equation becomes $\frac{d^2\theta}{dt^2}+A(\theta-\dfrac{\theta^3}{3!})=0 $ so , you can turn $$\frac{d^2\theta}{dt^2}+A\sin (\theta)=0 \\into \\\frac{d^2\theta}{dt^2}+A(1-\dfrac{\theta^2}{2!})=0$$ and solve numerically

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    If we wanted to solve this numerically why would we approximate it at all? Why not solve $\theta'' + A\cos\theta = 0$ directly? – 2017-01-31