# Derivation of Solow Growth Model (Solow, 1956)

In Solow's "A Contribution to the Theory of Economic Growth" paper, how do we go from this equation

to

such that

Could you please give me a hint about that derivation?

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We have that: $$\dot K = s K^\alpha L_0^b e^{nbt}$$ Rewriting the differential equation gives: $$K^{-\alpha} \frac{dK}{dt} = s L_0^b e^{nbt}$$ Integrate both sides with respect to $$t$$ from $$0$$ to $$T$$ gives: $$\frac{1}{b} [K^b]^T_0 = s L_0^b \frac{1}{nb}[e^{nbt}]^T_0$$ So: $$K^b_T = K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT}$$ Equivalently: $$K_T = \left[K^b_0 - \frac{s}{n} L_0^b + \frac{s}{n} L_0^b e^{nbT}\right]^{\frac{1}{b}}$$