How do you solve: $y'=c_1e^{-\frac{2}{3}x^{\frac{3}{2}}}$? How do you solve that ODE? I understand it has gamma functions but I have no clue where to start. Thanks
$$y'=c_1e^{-\frac{2}{3}x^{\frac{3}{2}}}$$
As Amzoti commented, just write $$y=c_1 \int e^{-\frac{2}{3}x^{\frac{3}{2}}}dx$$ Life could be easier if you change variable and do $$\frac{2}{3}x^{\frac{3}{2}}=z$$ $$x=\left(\frac{3}{2}\right)^{2/3} z^{2/3}$$ $$dx=\frac{\sqrt[3]{\frac{2}{3}}}{\sqrt[3]{z}}dz$$ which then leads to $$y=\sqrt[3]{\frac{2}{3}}c_1\int \frac{ e^{-z}}{\sqrt[3]{z}}dz$$

Since you mention gamma function, I am sure that you can take from here.
You can start by finding that $$\begin{array}{rrcl} &amp; c_{1}\cdot \mathsf{e}^{-\frac{2}{3}\cdot\sqrt{x^{3}}} &amp; = &amp; y^{\prime} \\ &amp; &amp; = &amp; \frac{\mathrm{d}y}{\mathrm{d}x} \\ \Rightarrow &amp; \mathrm{d}y &amp; = &amp; c_{1}\cdot\mathsf{e}^{-\frac{2}{3}\cdot\sqrt{x^{3}}}\,\mathrm{d}x \\ \Rightarrow &amp; y &amp; = &amp; \int c_{1}\cdot\mathsf{e}^{-\frac{2}{3}\cdot\sqrt{x^{3}}}\,\mathrm{d}x \end{array}$$