Internal problem ID [10737]
Internal file name [OUTPUT/9685_Monday_June_06_2022_03_22_01_PM_52889186/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 1.3.3-2. Equations
of the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 1.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\left (a x +3 b \right ) y=-a b \,x^{2}+c \,x^{3}-2 b^{2} x} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\left (a x +3 b \right ) y=-a b \,x^{2}+c \,x^{3}-2 b^{2} x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (a x +3 b \right ) y+c \,x^{3}-a b \,x^{2}-2 b^{2} x}{y} \end {array} \]
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 Looking for potential symmetries found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 224
dsolve(y(x)*diff(y(x),x)=(a*x+3*b)*y(x)+c*x^3-a*b*x^2-2*b^2*x,y(x), singsol=all)
\[ \frac {x \left (\frac {-2 y \left (x \right )^{2}+x \left (a x +4 b \right ) y \left (x \right )-a b \,x^{3}+c \,x^{4}-2 b^{2} x^{2}}{\left (b x -y \left (x \right )\right )^{2}}\right )^{\frac {1}{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {-a b x +2 c \,x^{2}+a y \left (x \right )}{\left (-b x +y \left (x \right )\right ) \sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}} y \left (x \right )+\sqrt {\frac {x^{2}}{-b x +y \left (x \right )}}\, \left (b x -y \left (x \right )\right ) \left (\left (\int _{}^{\frac {x^{2}}{-b x +y \left (x \right )}}\frac {\left (\textit {\_a}^{2} c +a \textit {\_a} -2\right )^{\frac {1}{4}} {\mathrm e}^{-\frac {a \,\operatorname {arctanh}\left (\frac {2 c \textit {\_a} +a}{\sqrt {a^{2}+8 c}}\right )}{2 \sqrt {a^{2}+8 c}}}}{\sqrt {\textit {\_a}}}d \textit {\_a} \right ) b +c_{1} \right )}{\sqrt {\frac {x^{2}}{-b x +y \left (x \right )}}\, \left (b x -y \left (x \right )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==(a*x+3*b)*y[x]+c*x^3-a*b*x^2-2*b^2*x,y[x],x,IncludeSingularSolutions -> True]
Not solved