Internal problem ID [10741]
Internal file name [OUTPUT/9689_Monday_June_06_2022_03_22_16_PM_26947723/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: 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
[_rational, [_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }+x \left (a \,x^{2}+b \right ) y=-x} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }+x \left (a \,x^{2}+b \right ) y=-x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {x \left (a \,x^{2}+b \right ) y+x}{y} \end {array} \]
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: 179
dsolve(y(x)*diff(y(x),x)+x*(a*x^2+b)*y(x)+x=0,y(x), singsol=all)
\[ \frac {2 \operatorname {AiryBi}\left (1, \frac {4 a y \left (x \right )+\left (a \,x^{2}+b \right )^{2}}{4 a^{\frac {2}{3}}}\right ) a^{\frac {1}{3}} c_{1} +c_{1} \left (a \,x^{2}+b \right ) \operatorname {AiryBi}\left (\frac {4 a y \left (x \right )+\left (a \,x^{2}+b \right )^{2}}{4 a^{\frac {2}{3}}}\right )-2 \operatorname {AiryAi}\left (1, \frac {4 a y \left (x \right )+\left (a \,x^{2}+b \right )^{2}}{4 a^{\frac {2}{3}}}\right ) a^{\frac {1}{3}}-\left (a \,x^{2}+b \right ) \operatorname {AiryAi}\left (\frac {4 a y \left (x \right )+\left (a \,x^{2}+b \right )^{2}}{4 a^{\frac {2}{3}}}\right )}{2 \operatorname {AiryBi}\left (1, \frac {4 a y \left (x \right )+\left (a \,x^{2}+b \right )^{2}}{4 a^{\frac {2}{3}}}\right ) a^{\frac {1}{3}}+\operatorname {AiryBi}\left (\frac {4 a y \left (x \right )+\left (a \,x^{2}+b \right )^{2}}{4 a^{\frac {2}{3}}}\right ) \left (a \,x^{2}+b \right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.492 (sec). Leaf size: 143
DSolve[y[x]*y'[x]+x*(a*x^2+b)*y[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\left (a x^2+b\right ) \operatorname {AiryAi}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \operatorname {AiryAiPrime}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}{\left (a x^2+b\right ) \operatorname {AiryBi}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \operatorname {AiryBiPrime}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}+c_1=0,y(x)\right ] \]