I am trying to solve a linear differential equation for the change in scale factor term, $\delta a(t)$, when mass terms are introduced to the equations of motion for General Relativity. The first equation I am have to solve is as follows:

$$ \frac{d}{dt} \delta a(t) = \frac{8 \pi G (a(t))^2}{3 \frac{d}{dt}a(t)}[3m^2 (a(t))^2 \mp \frac{1}{2} \delta \rho (t)] + \frac{2\frac{d}{dt}a(t) \delta a(t)}{a} $$ This equation is of the form $\frac{d}{dt} \delta a(t) + f(t) \delta a(t) = g(t)$, which is a linear differential equation.

However, when I plug this equation into Mathematica, using the DSolve function, I receive an error to the degree of

Equation or list of equations expected instead of True in the first argument

My code is as follows:

DSolve[{\[Delta]a'[t] - (2*a'[t]*\[Delta]a[t])/a[t] == (
    3*a'[t]) (3*m^2*(a^2)[t] - 1/2 \[Delta]\[Rho][t]), \[Delta]a[0] ==
    0, a[0] == 1}, \[Delta]a[t], t]

where I have introduced the conditions that $\delta a(0) = 0$ and $a(0) = 1$.

What am I doing wrong here, is this just not numerically solvable?



Change (a^2)[t] to a[t]^2 and remove the condition a[0] == 1 (NDSolve only accepts initial condition for \[Delta]a[t])

DSolve[{\[Delta]a'[t] - (2*a'[t]*\[Delta]a[t])/a[t] == (8*\[Pi]*G*a[t]^2)/(3*a'[t]) (3*m^2*a[t]^2 - 1/2 \[Delta]\[Rho][t]), \[Delta]a[0] == 0 }, \[Delta]a[t], t]
(*{{\[Delta]a[t] -> a[t]^2 Inactive[Integrate][(2 (12 G m^2 \[Pi] a[K[1]]^5 - 2 G \[Pi] a[K[1]]^3 \[Delta]\[Rho][K[1]]))/(3 a[K[1]]^3 Derivative[1][a][K[1]]), {K[1], 0, t}]}}*)
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