Internal problem ID [10681]
Internal file name [OUTPUT/9629_Monday_June_06_2022_03_16_46_PM_8383073/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. Form \(y y'-y=f(x)\). subsection 1.3.1-2.
Solvable equations and their solutions
Problem number: 33.
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 B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-y=\frac {A}{x^{2}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y y^{\prime } x^{2}+x^{2} y+A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {x^{2} y+A}{x^{2} y} \end {array} \]
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: 279
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(-2),y(x), singsol=all)
\[ \frac {-\left (\operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right ) c_{1} -\operatorname {AiryAi}\left (-\frac {\left (x^{3}-2 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right )\right ) A \left (-y \left (x \right )+x \right ) 2^{\frac {1}{3}}+2 \left (-A^{2}\right )^{\frac {2}{3}} \left (\operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right ) c_{1} -\operatorname {AiryAi}\left (1, -\frac {\left (x^{3}-2 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right )\right )}{-A 2^{\frac {1}{3}} \left (-y \left (x \right )+x \right ) \operatorname {AiryBi}\left (-\frac {\left (x^{3}-2 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right )+2 \operatorname {AiryBi}\left (1, -\frac {\left (x^{3}-2 y \left (x \right ) x^{2}+y \left (x \right )^{2} x +2 A \right ) 2^{\frac {2}{3}}}{4 \left (-A^{2}\right )^{\frac {1}{3}} x}\right ) \left (-A^{2}\right )^{\frac {2}{3}}} = 0 \]
✓ Solution by Mathematica
Time used: 1.053 (sec). Leaf size: 201
DSolve[y[x]*y'[x]-y[x]==A*x^(-2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\operatorname {AiryAiPrime}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )-\frac {(x-y(x)) \operatorname {AiryAi}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )}{2^{2/3} \sqrt [3]{A}}}{\operatorname {AiryBiPrime}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )-\frac {(x-y(x)) \operatorname {AiryBi}\left (\frac {x^3-2 y(x) x^2+y(x)^2 x+2 A}{2 \sqrt [3]{2} A^{2/3} x}\right )}{2^{2/3} \sqrt [3]{A}}}+c_1=0,y(x)\right ] \]