2.38   ODE No. 38

1. Problem in Latex
2. Mathematica input
3. Maple input

$$-ay(x{)}^3-\frac{b}{x^{3/2}}+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.160829 (sec), leaf count = 99

$$\text{Solve}[-2\text{RootSum}[-2{\mathrm{\#}\text{1}}^3+\mathrm{\#}\text{1}\sqrt[3]{-\frac{1}{a{b}^2}}-2\mathrm{\&},\frac{\log (y(x)\sqrt[3]{\frac{a{x}^{3/2}}{b}}-\mathrm{\#}\text{1})}{\sqrt[3]{-\frac{1}{a{b}^2}}-6{\mathrm{\#}\text{1}}^2}\mathrm{\&}]=\frac{ax\log (x)}{{\left(\frac{a{x}^{3/2}}{b}\right)}^{2/3}}+c_1,y(x)]$$

✓ Maple : cpu = 0.068 (sec), leaf count = 34

$$\{y\left(x\right)=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-\ln \left(x\right)+\mathit{\_}\mathit{C}\mathit{1}+2{\int }^{\mathit{\_}\mathit{Z}}{(2a{\mathit{\_}\mathit{a}}^3+\mathit{\_}\mathit{a}+2b)}^{-1}d\mathit{\_}\mathit{a})\frac{1}{\sqrt{x}}\}$$

Hand solution

$$\begin{array}{}\text{(1)}& y^{\prime }\left(x\right)=a{y}^3+b{x}^{-\frac{3}{2}}\end{array}$$

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

$$\begin{array}{rl}\frac{dy}{dx}& =-\frac{1}{2}{x}^{-\frac{3}{2}}\eta \left(\xi \right)+x^{-\frac{1}{2}}\frac{d\eta }{d\xi }\frac{d\xi }{dx}\\ & =-\frac{1}{2}{x}^{-\frac{3}{2}}\eta \left(\xi \right)+x^{-\frac{1}{2}}\frac{d\eta }{d\xi }\frac{1}{x}\\ & =-\frac{1}{2}{x}^{-\frac{3}{2}}\eta \left(\xi \right)+x^{-\frac{3}{2}}\frac{d\eta }{d\xi }\end{array}$$

Substituting in (1) gives

$$\begin{array}{rl}-\frac{1}{2}{x}^{-\frac{3}{2}}\eta \left(\xi \right)+x^{-\frac{3}{2}}\frac{d\eta }{d\xi }& =a{\left(x^{-\frac{1}{2}}\eta \left(\xi \right)\right)}^3+b{x}^{-\frac{3}{2}}\\ -\frac{1}{2}{x}^{-\frac{3}{2}}\eta \left(\xi \right)+x^{-\frac{3}{2}}\frac{d\eta }{d\xi }& =a{x}^{-\frac{3}{2}}{\eta }^3\left(\xi \right)+b{x}^{-\frac{3}{2}}\\ -\frac{1}{2}\eta +{\eta }^{\prime }& =a{\eta }^3+b\\ {\eta }^{\prime }& =b+\frac{1}{2}\eta +a{\eta }^3\end{array}$$

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

$$\eta =\xi -{\int }^{\eta \left(\xi \right)}\frac{1}{b+\frac{1}{2}z+a{z}^3}dz+C$$

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

$$\begin{array}{rl}\sqrt{x}y& =\ln x-{\int }^{\sqrt{x}y}\frac{1}{b+\frac{1}{2}z+a{z}^3}dz+C\\ y\left(x\right)& =(\ln x-{\int }^{\sqrt{x}y}\frac{1}{b+\frac{1}{2}z+a{z}^3}dz+C)\frac{1}{\sqrt{x}}\end{array}$$

DId not verify. Need to look more into this later.