Your solution seems fine. What does the rest of the error message say?
![\displaystyle y^{1/2}\frac{\mathrm dy}{\mathrm dx} + y^{3/2} = 1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%5E%7B1%2F2%7D%5Cfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D%20%2B%20y%5E%7B3%2F2%7D%20%3D%201)
Substitute
![z(x)=y(x)^{3/2} \implies \dfrac{\mathrm dz}{\mathrm dx}=\dfrac32y(x)^{1/2}\dfrac{\mathrm dy}{\mathrm dx}](https://tex.z-dn.net/?f=z%28x%29%3Dy%28x%29%5E%7B3%2F2%7D%20%5Cimplies%20%5Cdfrac%7B%5Cmathrm%20dz%7D%7B%5Cmathrm%20dx%7D%3D%5Cdfrac32y%28x%29%5E%7B1%2F2%7D%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dx%7D)
to transform the ODE to a linear one in <em>z</em> :
![\displaystyle \frac23\frac{\mathrm dz}{\mathrm dx} + z = 1](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac23%5Cfrac%7B%5Cmathrm%20dz%7D%7B%5Cmathrm%20dx%7D%20%2B%20z%20%3D%201)
Divide both sides by 2/3 :
![\displaystyle \frac{\mathrm dz}{\mathrm dx} + \frac32z = \frac32](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cmathrm%20dz%7D%7B%5Cmathrm%20dx%7D%20%2B%20%5Cfrac32z%20%3D%20%5Cfrac32)
Multiply both sides by the integrating factor,
:
![\displaystyle e^{3x/2}\frac{\mathrm dz}{\mathrm dx} + \frac32 e^{3x/2}z = \frac32 e^{3x/2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20e%5E%7B3x%2F2%7D%5Cfrac%7B%5Cmathrm%20dz%7D%7B%5Cmathrm%20dx%7D%20%2B%20%5Cfrac32%20e%5E%7B3x%2F2%7Dz%20%3D%20%5Cfrac32%20e%5E%7B3x%2F2%7D)
Condense the left side into the derivative of a product :
![\displaystyle \frac{\mathrm d}{\mathrm dx}\left[e^{3x/2}z\right] = \frac32 e^{3x/2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5Be%5E%7B3x%2F2%7Dz%5Cright%5D%20%3D%20%5Cfrac32%20e%5E%7B3x%2F2%7D)
Integrate both sides and solve for <em>z</em> :
![\displaystyle e^{3x/2}z = \frac32 \int e^{3x/2}\,\mathrm dx \\\\ e^{3x/2}z = e^{3x/2} + C \\\\ z = 1 + Ce^{-3x/2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20e%5E%7B3x%2F2%7Dz%20%3D%20%5Cfrac32%20%5Cint%20e%5E%7B3x%2F2%7D%5C%2C%5Cmathrm%20dx%20%5C%5C%5C%5C%20e%5E%7B3x%2F2%7Dz%20%3D%20e%5E%7B3x%2F2%7D%20%2B%20C%20%5C%5C%5C%5C%20z%20%3D%201%20%2B%20Ce%5E%7B-3x%2F2%7D)
Solve in terms of <em>y</em> :
![y^{3/2} = 1 + Ce^{-3x/2}](https://tex.z-dn.net/?f=y%5E%7B3%2F2%7D%20%3D%201%20%2B%20Ce%5E%7B-3x%2F2%7D)
Given that <em>y</em> (0) = 16, we have
![16^{3/2} = 1 + Ce^0 \implies C = 16^{3/2}-1 = 63](https://tex.z-dn.net/?f=16%5E%7B3%2F2%7D%20%3D%201%20%2B%20Ce%5E0%20%5Cimplies%20C%20%3D%2016%5E%7B3%2F2%7D-1%20%3D%2063)
so that the particular solution is
![\boxed{y^{3/2} = 1 + 63e^{-3x/2}}](https://tex.z-dn.net/?f=%5Cboxed%7By%5E%7B3%2F2%7D%20%3D%201%20%2B%2063e%5E%7B-3x%2F2%7D%7D)