Reducing a Second-Order ODE

Suppose we have the specific second-order equation: $\frac{d^{2}y}{d{t}^2}+\frac{dy}{dt}=(1+4t)\sqrt{y}$.

First, isolate the highest derivative: $\frac{d^{2}y}{d{t}^2}=(1+4t)\sqrt{y}-\frac{dy}{dt}$.

Introduce $z_1=y$ and $z_2=y^{\prime }$. The reduced first-order system is: $z_1^{\prime }=z_2$ and $z_2^{\prime }=(1+4t)\sqrt{z_1}-z_2$.

We can now apply standard methods like Forward Euler to this coupled system simultaneously.