scaling equation within margin

scaling equation within margin

I have split an equation across several lines but I couldn't remain
perfectly within the margin.

The input is
\begin{equation*}
\begin{split}
& \ddot\phi_3+\phi_3+sin(\tau+\alpha)[p_1\Delta \alpha+2 \nabla p_1
\nabla \alpha] \\
& \qquad -\cos(\tau+ \alpha)[\Delta p_1 -p_1 (\nabla
\alpha)^2+\frac{5}{6}g_2^2p_1^3- \frac{3}{4}g_3p_1^3-p_1 +\omega_2
p_1]
+\frac{ p_1^3}{12}(2g_2^2+3g_3)\cos3(\tau + \alpha) \\
& \qquad +g_2 p_1 [p_2+ p_2 cos2(\tau +\alpha)+q_2 \sin2(\tau+\alpha)] =0
\end{split}
\end{equation*}
\begin{equation}\label{eq10}
\begin{split}
& \ddot\phi_3+\phi_3+\sin(\tau+\alpha)[p_1\Delta \alpha+2 \nabla p_1
\nabla \alpha] \\
& \qquad -\cos(\tau+ \alpha)[\Delta p_1-p_1 (\nabla \alpha)^2 + \lambda
p_1^3-p_1+\omega_2 p_1] +\frac{p_1^3}{12}(2g_2^2+3g_3)\cos3(\tau + \alpha)
\\
& \qquad +g_2 p_1 [p_2+ p_2 \cos2(\tau +\alpha)+q_2 \sin2(\tau+\alpha)]=0
\end{split}
\end{equation}