24.5 problem 5

Internal problem ID [10752]

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`]]

\[ \boxed {y y^{\prime }+x \left (a \,x^{2}+b \right ) y=-x} \]

Solution by Maple

Time used: 0.0 (sec). Leaf size: 157

dsolve(y(x)*diff(y(x),x)+x*(a*x^2+b)*y(x)+x=0,y(x), singsol=all)
 

\[ c_{1} +\frac {-2 \operatorname {AiryAi}\left (1, \frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \left (x \right )+b^{2}}{4 a^{\frac {2}{3}}}\right ) a^{\frac {1}{3}}+\left (-a \,x^{2}-b \right ) \operatorname {AiryAi}\left (\frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \left (x \right )+b^{2}}{4 a^{\frac {2}{3}}}\right )}{\left (a \,x^{2}+b \right ) \operatorname {AiryBi}\left (\frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \left (x \right )+b^{2}}{4 a^{\frac {2}{3}}}\right )+2 \operatorname {AiryBi}\left (1, \frac {a^{2} x^{4}+2 a b \,x^{2}+4 a y \left (x \right )+b^{2}}{4 a^{\frac {2}{3}}}\right ) a^{\frac {1}{3}}} = 0 \]

Solution by Mathematica

Time used: 0.492 (sec). Leaf size: 143

DSolve[y[x]*y'[x]+x*(a*x^2+b)*y[x]+x==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {\left (a x^2+b\right ) \operatorname {AiryAi}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \operatorname {AiryAiPrime}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}{\left (a x^2+b\right ) \operatorname {AiryBi}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )+2 \sqrt [3]{a} \operatorname {AiryBiPrime}\left (\frac {\left (a x^2+b\right )^2+4 a y(x)}{4 a^{2/3}}\right )}+c_1=0,y(x)\right ] \]