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Is it allowed to use the quadratic solution formula for a differential equation

I have some trouble with a challenging fluid mechanics problem. The problem leads me to a non-linear ode 1st order. $0={\dot p_C}^2+\frac{k_1}{k_2 C}\dot p_C+\frac{p_C-p_0}{k_2C^2}$ My Idea was now ...