22.23 problem 23

Internal problem ID [9925]

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: 23.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class B`]]

\[ \boxed {y^{\prime } y-y+\frac {9 x}{100}-\frac {A}{x^{\frac {5}{3}}}=0} \]

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 551

dsolve(y(x)*diff(y(x),x)-y(x)=-9/100*x+A*x^(-5/3),y(x), singsol=all)
 

\[ c_{1} +\frac {10460353203000 A \,x^{\frac {44}{3}}-895371796800000 A \,x^{\frac {35}{3}} y \left (x \right )^{3}+1205308188000000 A \,x^{\frac {32}{3}} y \left (x \right )^{4}-892820880000000 A \,x^{\frac {29}{3}} y \left (x \right )^{5}[phone] A \,x^{\frac {26}{3}} y \left (x \right )^{6}[phone] A \,x^{\frac {23}{3}} y \left (x \right )^{7}+1629752400000000 A^{2} x^{9} y \left (x \right )^{3}-531441000000000 A^{2} x^{8} y \left (x \right )^{4}[phone] A^{2} x^{7} y \left (x \right )^{5}-417960000000000 A^{2} x^{6} y \left (x \right )^{6}[phone] A^{2} x^{5} y \left (x \right )^{7}[phone] A^{3} x^{\frac {13}{3}} y \left (x \right )^{5}-291600000000000 A^{4} x^{\frac {11}{3}} y \left (x \right )^{3}-97629963228000 A \,x^{\frac {41}{3}} y \left (x \right )+1968300000000000 A^{5} x^{4}-135000000000000 A^{2} x^{4} y \left (x \right )^{8}-540000000000000 A^{3} x^{\frac {10}{3}} y \left (x \right )^{6}-1215000000000000 A^{4} x^{\frac {8}{3}} y \left (x \right )^{4}-1930902300000000 A \left (\frac {1400 y \left (x \right )^{8}}{8829}+A^{3}\right ) x^{\frac {20}{3}}+1614006000000000 y \left (x \right )^{2} \left (-\frac {110 y \left (x \right )^{8}}{3321}+A^{3}\right ) x^{\frac {22}{3}}-2111592240000000 y \left (x \right ) \left (-\frac {2750 y \left (x \right )^{8}}{36207}+A^{3}\right ) x^{\frac {25}{3}}[phone] y \left (x \right )^{4} \left (-\frac {25 y \left (x \right )^{8}}{5103}+A^{3}\right ) x^{\frac {16}{3}}-571536000000000 \left (-\frac {25 y \left (x \right )^{8}}{1323}+A^{3}\right ) y \left (x \right )^{3} x^{\frac {19}{3}}[phone] A \,x^{\frac {38}{3}} y \left (x \right )^{2}+1093500000000000 y \left (x \right )^{2} A \left (-\frac {4 y \left (x \right )^{8}}{243}+A^{3}\right ) x^{\frac {14}{3}}+2283228000000000 y \left (x \right ) \left (\frac {1100 y \left (x \right )^{8}}{21141}+A^{3}\right ) A \,x^{\frac {17}{3}}-140762777670000 A^{2} x^{12}-282429536481 x^{\frac {52}{3}}-729000000000000 A^{6} x^{\frac {4}{3}}[phone] x^{\frac {49}{3}} y \left (x \right )-23012777046600 x^{\frac {46}{3}} y \left (x \right )^{2}[phone] x^{\frac {43}{3}} y \left (x \right )^{3}-213081268950000 x^{\frac {40}{3}} y \left (x \right )^{4}[phone] x^{\frac {37}{3}} y \left (x \right )^{5}-491051484000000 x^{\frac {34}{3}} y \left (x \right )^{6}[phone] x^{\frac {31}{3}} y \left (x \right )^{7}+\left (-1702736964000000 A^{2} x^{10}-1458000000000000 A^{5} x^{2}\right ) y \left (x \right )^{2}+\left (792059666400000 A^{2} x^{11}-874800000000000 A^{5} x^{3}\right ) y \left (x \right )+\left (-324769500000000 y \left (x \right )^{8}[phone] A^{3}\right ) x^{\frac {28}{3}}}{729 x^{\frac {28}{3}}-9720 x^{\frac {25}{3}} y \left (x \right )+27000 x^{\frac {22}{3}} y \left (x \right )^{2}-28000 x^{\frac {19}{3}} y \left (x \right )^{3}+10000 y \left (x \right )^{4} x^{\frac {16}{3}}-16200 A \,x^{\frac {20}{3}}+108000 A \,x^{\frac {17}{3}} y \left (x \right )+60000 A \,x^{\frac {14}{3}} y \left (x \right )^{2}+90000 A^{2} x^{4}} = 0 \]

✓ Solution by Mathematica

Time used: 60.339 (sec). Leaf size: 7909

DSolve[y[x]*y'[x]-y[x]==-9/100*x+A*x^(-5/3),y[x],x,IncludeSingularSolutions -> True]
 

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