2.38   ODE No. 38

$-a y(x)^3-\frac {b}{x^{3/2}}+y'(x)=0$ Mathematica : cpu = 0.155898 (sec), leaf count = 99

$\text {Solve}\left [-2 \text {RootSum}\left [-2 \text {\#1}^3+\text {\#1} \sqrt [3]{-\frac {1}{a b^2}}-2\& ,\frac {\log \left (y(x) \sqrt [3]{\frac {a x^{3/2}}{b}}-\text {\#1}\right )}{\sqrt [3]{-\frac {1}{a b^2}}-6 \text {\#1}^2}\& \right ]=\frac {a x \log (x)}{\left (\frac {a x^{3/2}}{b}\right )^{2/3}}+c_1,y(x)\right ]$ Maple : cpu = 0.022 (sec), leaf count = 34

$\left \{ y \left ( x \right ) ={{\it RootOf} \left ( -\ln \left ( x \right ) +{\it \_C1}+2\,\int ^{{\it \_Z}}\! \left ( 2\,a{{\it \_a}}^{3}+{\it \_a}+2\,b \right ) ^{-1}{d{\it \_a}} \right ) {\frac {1}{\sqrt {x}}}} \right \}$

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.