How do I solve a linear ODE system in $n+1$ dimensions, i.e. $∂y/∂t+p(t)y=q(x_1,….,x_n,t)$ Where $y=y(x_1,….,x_n,t)$ and initial conditions are $y_0=y(x_1 (0),….,x_n (0),0)$? I am familiar with use of integrating factor in one dimension, but can’t adapt it.
Integrate p(s) from 0 to t. Then put this in the exponent and you will have an integrating factor. Multiplying both sides by this will give you the partial derivative of the product (Integrating Factor x p(t)) with respect to t on the left hand side. You can integrate this over t and the right hand side over t and you will get a solution. You can use the initial conditions to determine the bounds of integration.