A key solution addresses the "scale effect" prediction (larger populations yield faster growth). Modern solutions show how to modify the model to eliminate this unrealistic prediction by introducing diminishing returns to the stock of ideas.
The authors use differential equations to find the point where an economy’s capital stock stays constant. They prove that in the long run, the growth rate of output per worker depends entirely on the rate of technological progress. Convergence Analysis
If India has β=0.02 and initial GDP 1/4 of the US, the solution predicts India will close half the gap in ln ~35 years.