\[ -a y(x)^3-\frac {b}{x^{3/2}}+y'(x)=0 \] ✓ Mathematica : cpu = 0.31404 (sec), leaf count = 320
DSolve[-(b/x^(3/2)) - a*y[x]^3 + Derivative[1][y][x] == 0,y[x],x]
\[\text {Solve}\left [\frac {2}{3} a b^2 \text {RootSum}\left [8 \text {$\#$1}^9 a b^2+24 \text {$\#$1}^6 a b^2+24 \text {$\#$1}^3 a b^2+\text {$\#$1}^3+8 a b^2\& ,\frac {4 \text {$\#$1}^6 \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+2 \text {$\#$1}^4 \sqrt [3]{-\frac {1}{a b^2}} \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+8 \text {$\#$1}^3 \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+\text {$\#$1}^2 \left (-\frac {1}{a b^2}\right )^{2/3} \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+2 \text {$\#$1} \sqrt [3]{-\frac {1}{a b^2}} \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )+4 \log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {$\#$1}\right )}{24 \text {$\#$1}^8 a b^2+48 \text {$\#$1}^5 a b^2+24 \text {$\#$1}^2 a b^2+\text {$\#$1}^2}\& \right ]=\frac {a x \log (x)}{\left (\frac {a x^{3/2}}{b}\right )^{2/3}}+c_1,y(x)\right ]\] ✓ Maple : cpu = 0.02 (sec), leaf count = 34
dsolve(diff(y(x),x)-a*y(x)^3-b/x^(3/2) = 0,y(x))
\[y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1}+2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 a \,\textit {\_a}^{3}+\textit {\_a} +2 b}d \textit {\_a} \right )\right )}{\sqrt {x}}\]
Hand solution
\begin {equation} y^{\prime }\left ( x\right ) =ay^{3}+bx^{-\frac {3}{2}}\tag {1} \end {equation}
This can be transformed to Abel first order non-linear ode as follows. Let \(y\left ( x\right ) =x^{-\frac {1}{2}}\eta \left ( \xi \right ) \) where \(\xi =\ln x\) hence
\begin {align*} \frac {dy}{dx} & =-\frac {1}{2}x^{-\frac {3}{2}}\eta \left ( \xi \right ) +x^{-\frac {1}{2}}\frac {d\eta }{d\xi }\frac {d\xi }{dx}\\ & =-\frac {1}{2}x^{-\frac {3}{2}}\eta \left ( \xi \right ) +x^{-\frac {1}{2}}\frac {d\eta }{d\xi }\frac {1}{x}\\ & =-\frac {1}{2}x^{-\frac {3}{2}}\eta \left ( \xi \right ) +x^{-\frac {3}{2}}\frac {d\eta }{d\xi } \end {align*}
Substituting in (1) gives
\begin {align*} -\frac {1}{2}x^{-\frac {3}{2}}\eta \left ( \xi \right ) +x^{-\frac {3}{2}}\frac {d\eta }{d\xi } & =a\left ( x^{-\frac {1}{2}}\eta \left ( \xi \right ) \right ) ^{3}+bx^{-\frac {3}{2}}\\ -\frac {1}{2}x^{-\frac {3}{2}}\eta \left ( \xi \right ) +x^{-\frac {3}{2}}\frac {d\eta }{d\xi } & =ax^{-\frac {3}{2}}\eta ^{3}\left ( \xi \right ) +bx^{-\frac {3}{2}}\\ -\frac {1}{2}\eta +\eta ^{\prime } & =a\eta ^{3}+b\\ \eta ^{\prime } & =b+\frac {1}{2}\eta +a\eta ^{3} \end {align*}
This is Abel first kind. In general form it is
\[ \eta ^{\prime }=f_{0}+f_{1}\eta +f_{2}\eta ^{2}+f_{3}\eta ^{3}\]
Where in this case \(f_{0}=b,f_{1}=\frac {1}{2},f_{2}=0,f_{3}=a\). Using Maple, the solution to the above is (I need to learn how to solve Able by hand more) is implicit, given as
\[ \eta =\xi -\int ^{\eta \left ( \xi \right ) }\frac {1}{b+\frac {1}{2}z+az^{3}}dz+C \]
Where \(C\) is constant of integration. Hence, since \(y\left ( x\right ) =x^{-\frac {1}{2}}\eta \left ( \xi \right ) \), then \(\eta \left ( \xi \right ) =\sqrt {x}y\) and the above becomes
\begin {align*} \sqrt {x}y & =\ln x-\int ^{\sqrt {x}y}\frac {1}{b+\frac {1}{2}z+az^{3}}dz+C\\ y\left ( x\right ) & =\left ( \ln x-\int ^{\sqrt {x}y}\frac {1}{b+\frac {1}{2}z+az^{3}}dz+C\right ) \frac {1}{\sqrt {x}} \end {align*}
DId not verify. Need to look more into this later.