I’m doing first order ODE questions using lie symmetries. This is a made up question:  Dy/dx = x + 2y = w(x, y). the point of this question was a way for me to learn the steps to solve a first order equation using lie symmetries and learn to make reasonable assumptions.   For the question above, I assume zeta = 0 and eta = eta(x).  my final answer is y = 2x + (x^2). is that correct? I’ll show my working out below.   I worked out eta as eta= (exp^(2x + c1)) -> after subbing in my assumptions eta = eta(x) and zeta = 0. I subbed this into the canonical variables equations to get: r=x, s= y/(exp^(2x + c1)).   I worked out dr/dx = 1, dr/dy = zero, ds/dy = 1/(exp^(2x + c1)) and ds/dx = (2-2y)/ (exp^(2x + c1)). Omega(x,y) is given in the question. I now do ds/dr using the formula: [ds/dx + omega(x,y)ds/dy] / [ dr/dx + omega(x,y)dr/dy].   when I plug everything into the ds/dr formula, I get: ((2-2y)/ (exp^(2x + c1))) + (x + 2y)(1/ exp^(2x + c1)) / (1 + (x + 2y)(0). the denominator becomes 1, and I expand the numerator: ((2-2y) / exp^(2x + c1)) + (x / exp^(2x + c1)) + (2y / exp^(2x + c1)). then the 2y cancels out on top to give ((2 + x) / exp^(2x + c1)).   Now that I have ds/dr = (((2 + x) / exp^(2x + c1)), I take dr to the other side and integrate: integral (ds) = integral ((2 + x) / exp^(2x + c1)) dr. This gives: S = ((2r + rx) / exp^(2x + c1))   I now sub in my r and s values: r=x and s= y / exp^(2x + c1), to transform back to x and y ->  y / (exp^(2x + c1)) = ((2x + x^2) / (exp^(2x + c1)).   I get the final answer of y = 2x + (x^2).   Is that correct?   Can you check my solution, y = 2x + (x^2), by substituting into the original equation, Dy/dx = x + 2y. when i did it is not working. could you go through my working out and tell me where i went wrong please?