I am trying to solve problem 1.20 in the book 'Physical Foundations of Cosmology' from Mukhanov.
In the problem, we should show that the scale factor $a$ for a flat universe filled with cosmological constant and radiation is given by, $$ a(t)=a_0(\sinh2H_\Lambda t)^{1/2},\tag{1} $$ wherein $H_\Lambda=(8\pi G\epsilon_\Lambda/3)^{1/2}$.
As a hint, we are given, $$ a^{\prime\prime}+ka=\frac{4\pi G}{3}\left(\epsilon-3p\right)a^3,\tag{2} $$ where prime denotes the derivative with respect to conformal time $\eta$.
The equation of state for the cosmological constant $\Lambda$ is $p=-\epsilon$ whereas the equation of state for radiation is $p=\frac{1}{3}\epsilon$. Using these equations the right-hand side of the former equation yields, $$ \frac{4\pi G}{3}4\epsilon_\Lambda a^3=2H_\Lambda a^3.\tag{3} $$ As for a flat universe $k=0$ we arrive at, $$ a^{\prime\prime}=2H_\Lambda a^3.\tag{4} $$
If we multiply both sides with $a^\prime$ we can write, $$ \frac{1}{2}\frac{d}{d\eta}(a^\prime)^2=a^{\prime\prime}a^\prime=2H_\Lambda a^3a^\prime=\frac{1}{2}H_\Lambda \frac{d}{d\eta}a^4,\tag{5} $$ and we can integrate both sides with respect to $\eta$.
Dropping the integration constant and taking the square root of both sides, $$ a^\prime=\pm H_\Lambda^{1/2}a^2,\tag{6} $$ we can solve this by separation and of variables and find, $$ \eta=\int d\eta=\pm H^{-1/2}_\Lambda\int \frac{da}{a^2}=\mp H^{-1/2}_\Lambda \frac{1}{a}.\tag{7} $$
We need the scale factor to be positive, therefore, $$ a(\eta)=H^{-1/2}_\Lambda \eta.\tag{8} $$
From the definition of the conformal time, $$ \eta=\int\frac{dt}{a(t)},\tag{9} $$ we find, $$ t=\int dt=\int d\eta a(\eta)=\frac{1}{2}H^{-1/2}_\Lambda \eta^2,\tag{10} $$ which solved for $\eta$ can be used to express the scale factor in proper time $t$, $$ a(t)=\sqrt{2 t}H_\Lambda^{3/4},\tag{11} $$ which obviously is very different from the actual result.
What did I wrong?