Internal problem ID [9983]
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 1.3.2. Equations of the
form \(y y'=f(x) y+1\)
Problem number: 1.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y y^{\prime }-\left (a x +b \right ) y-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 191
dsolve(y(x)*diff(y(x),x)=(a*x+b)*y(x)+1,y(x), singsol=all)
\[ c_{1}+\frac {-2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \left (x a +b \right ) \AiryAi \left (-\frac {2^{\frac {2}{3}} \left (a^{2} x^{2}+\left (2 b x -2 y \relax (x )\right ) a +b^{2}\right )}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right )-2 \AiryAi \left (1, -\frac {2^{\frac {2}{3}} \left (a^{2} x^{2}+\left (2 b x -2 y \relax (x )\right ) a +b^{2}\right )}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right ) a}{2^{\frac {1}{3}} \left (-a^{2}\right )^{\frac {1}{3}} \left (x a +b \right ) \AiryBi \left (-\frac {2^{\frac {2}{3}} \left (a^{2} x^{2}+\left (2 b x -2 y \relax (x )\right ) a +b^{2}\right )}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right )+2 \AiryBi \left (1, -\frac {2^{\frac {2}{3}} \left (a^{2} x^{2}+\left (2 b x -2 y \relax (x )\right ) a +b^{2}\right )}{4 \left (-a^{2}\right )^{\frac {1}{3}}}\right ) a} = 0 \]
✓ Solution by Mathematica
Time used: 0.535 (sec). Leaf size: 161
DSolve[y[x]*y'[x]==(a*x+b)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\sqrt [3]{2} (a x+b) \text {Ai}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \text {Ai}'\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}{\sqrt [3]{2} (a x+b) \text {Bi}\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )-2 \sqrt [3]{a} \text {Bi}'\left (\frac {(b+a x)^2-2 a y(x)}{2 \sqrt [3]{2} a^{2/3}}\right )}+c_1=0,y(x)\right ] \]