Internal problem ID [10680]
Internal file name [OUTPUT/9628_Monday_June_06_2022_03_16_45_PM_69846556/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: 32.
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.
\[ \boxed {y y^{\prime }-y=\frac {A}{\sqrt {x}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -y y^{\prime } \sqrt {x}+y \sqrt {x}+A =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y \sqrt {x}+A}{y \sqrt {x}} \end {array} \]
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: 0.015 (sec). Leaf size: 222
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(-1/2),y(x), singsol=all)
\[ \frac {\left (\operatorname {AiryBi}\left (-\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (y \left (x \right )-x \right )}{2 A^{2} x}\right ) c_{1} -\operatorname {AiryAi}\left (-\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (y \left (x \right )-x \right )}{2 A^{2} x}\right )\right ) 2^{\frac {2}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {1}{3}}-2 A \left (-\operatorname {AiryBi}\left (1, -\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (y \left (x \right )-x \right )}{2 A^{2} x}\right ) c_{1} +\operatorname {AiryAi}\left (1, -\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (y \left (x \right )-x \right )}{2 A^{2} x}\right )\right )}{2^{\frac {2}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {1}{3}} \operatorname {AiryBi}\left (-\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (y \left (x \right )-x \right )}{2 A^{2} x}\right )+2 \operatorname {AiryBi}\left (1, -\frac {2^{\frac {1}{3}} \left (-A^{2} x^{\frac {3}{2}}\right )^{\frac {2}{3}} \left (y \left (x \right )-x \right )}{2 A^{2} x}\right ) A} = 0 \]
✓ Solution by Mathematica
Time used: 0.566 (sec). Leaf size: 139
DSolve[y[x]*y'[x]-y[x]==A*x^(-1/2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [3]{-1} 2^{2/3} \sqrt {x} \operatorname {AiryAi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \operatorname {AiryAiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}{\sqrt [3]{-1} 2^{2/3} \sqrt {x} \operatorname {AiryBi}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )+2 \sqrt [3]{A} \operatorname {AiryBiPrime}\left (\frac {\left (-\frac {1}{2}\right )^{2/3} (x-y(x))}{A^{2/3}}\right )}+c_1=0,y(x)\right ] \]