Internal problem ID [9999]
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.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y y^{\prime }+x \left (a \,x^{2}+b \right ) y+x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 157
dsolve(y(x)*diff(y(x),x)+x*(a*x^2+b)*y(x)+x=0,y(x), singsol=all)
\[ c_{1}+\frac {-2 \AiryAi \left (1, \frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \relax (x )+b^{2}}{4 a^{\frac {2}{3}}}\right ) a^{\frac {1}{3}}+\left (-a \,x^{2}-b \right ) \AiryAi \left (\frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \relax (x )+b^{2}}{4 a^{\frac {2}{3}}}\right )}{2 \AiryBi \left (1, \frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \relax (x )+b^{2}}{4 a^{\frac {2}{3}}}\right ) a^{\frac {1}{3}}+\left (a \,x^{2}+b \right ) \AiryBi \left (\frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \relax (x )+b^{2}}{4 a^{\frac {2}{3}}}\right )} = 0 \]
✓ Solution by Mathematica
Time used: 0.31 (sec). Leaf size: 143
DSolve[y[x]*y'[x]+x*(a*x^2+b)*y[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {\left (a x^2+b\right ) \text {Ai}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \text {Ai}'\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}{\left (a x^2+b\right ) \text {Bi}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \text {Bi}'\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}+c_1=0,y(x)\right ] \]