Internal problem ID [10729]
Internal file name [OUTPUT/9677_Monday_June_06_2022_03_21_38_PM_47417038/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.2. Equations of
the form \(y y'=f(x) y+1\)
Problem number: 5.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\frac {3 y}{\sqrt {a \,x^{\frac {3}{2}}+8 x}}=1} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } \sqrt {a \,x^{\frac {3}{2}}+8 x}-3 y-\sqrt {a \,x^{\frac {3}{2}}+8 x}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {3 y+\sqrt {a \,x^{\frac {3}{2}}+8 x}}{\sqrt {a \,x^{\frac {3}{2}}+8 x}\, y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 293
dsolve(y(x)*diff(y(x),x)=3*(a*x^(3/2)+8*x)^(-1/2)*y(x)+1,y(x), singsol=all)
\[ \frac {{\left (-\frac {a \sqrt {x}\, \left (-2 a \,x^{\frac {3}{2}}+\sqrt {x}\, a y \left (x \right )^{2}-8 \sqrt {x \left (8+\sqrt {x}\, a \right )}\, y \left (x \right )-16 x \right )}{\left (\sqrt {x}\, a y \left (x \right )-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}\right )^{2}}\right )}^{\frac {1}{4}} \sqrt {2 \sqrt {x}\, a +16}\, a \sqrt {x}\, y \left (x \right )+4 \sqrt {-\frac {\sqrt {2 \sqrt {x}\, a +16}\, \sqrt {x \left (8+\sqrt {x}\, a \right )}}{\sqrt {x}\, a y \left (x \right )-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}}}\, \left (\sqrt {x}\, a y \left (x \right )-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}\right ) \left (\int _{}^{-\frac {\sqrt {2 \sqrt {x}\, a +16}\, \sqrt {x \left (8+\sqrt {x}\, a \right )}}{\sqrt {x}\, a y \left (x \right )-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}}}\frac {\left (\textit {\_a}^{2}-1\right )^{\frac {1}{4}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +\frac {c_{1}}{4}\right )}{\sqrt {-\frac {\sqrt {2 \sqrt {x}\, a +16}\, \sqrt {x \left (8+\sqrt {x}\, a \right )}}{\sqrt {x}\, a y \left (x \right )-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}}}\, \left (\sqrt {x}\, a y \left (x \right )-4 \sqrt {x \left (8+\sqrt {x}\, a \right )}\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==3*(a*x^(3/2)+8*x)^(-1/2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
Not solved