Internal problem ID [9980]
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: 2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {y^{\prime } y-\frac {y}{\left (a x +b \right )^{2}}-1=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 415
dsolve(y(x)*diff(y(x),x)=(a*x+b)^(-2)*y(x)+1,y(x), singsol=all)
\[ c_{1} +\frac {-a 2^{\frac {1}{3}} \left (y \left (x \right ) a^{2} x +y \left (x \right ) a b +1\right ) \operatorname {AiryAi}\left (-\frac {2^{\frac {2}{3}} \left (-\frac {1}{2}+x^{2} \left (-\frac {y \left (x \right )^{2}}{2}+x \right ) a^{4}+3 x b \left (-\frac {y \left (x \right )^{2}}{3}+x \right ) a^{3}+\left (\left (-\frac {y \left (x \right )^{2}}{2}+3 x \right ) b^{2}-x y \left (x \right )\right ) a^{2}+b \left (b^{2}-y \left (x \right )\right ) a \right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right )-2 \operatorname {AiryAi}\left (1, -\frac {2^{\frac {2}{3}} \left (-\frac {1}{2}+x^{2} \left (-\frac {y \left (x \right )^{2}}{2}+x \right ) a^{4}+3 x b \left (-\frac {y \left (x \right )^{2}}{3}+x \right ) a^{3}+\left (\left (-\frac {y \left (x \right )^{2}}{2}+3 x \right ) b^{2}-x y \left (x \right )\right ) a^{2}+b \left (b^{2}-y \left (x \right )\right ) a \right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right ) \left (a^{2}\right )^{\frac {2}{3}} \left (a x +b \right )}{a 2^{\frac {1}{3}} \left (y \left (x \right ) a^{2} x +y \left (x \right ) a b +1\right ) \operatorname {AiryBi}\left (-\frac {2^{\frac {2}{3}} \left (-\frac {1}{2}+x^{2} \left (-\frac {y \left (x \right )^{2}}{2}+x \right ) a^{4}+3 x b \left (-\frac {y \left (x \right )^{2}}{3}+x \right ) a^{3}+\left (\left (-\frac {y \left (x \right )^{2}}{2}+3 x \right ) b^{2}-x y \left (x \right )\right ) a^{2}+b \left (b^{2}-y \left (x \right )\right ) a \right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right )+2 \operatorname {AiryBi}\left (1, -\frac {2^{\frac {2}{3}} \left (-\frac {1}{2}+x^{2} \left (-\frac {y \left (x \right )^{2}}{2}+x \right ) a^{4}+3 x b \left (-\frac {y \left (x \right )^{2}}{3}+x \right ) a^{3}+\left (\left (-\frac {y \left (x \right )^{2}}{2}+3 x \right ) b^{2}-x y \left (x \right )\right ) a^{2}+b \left (b^{2}-y \left (x \right )\right ) a \right )}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (a x +b \right )^{2}}\right ) \left (a^{2}\right )^{\frac {2}{3}} \left (a x +b \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.367 (sec). Leaf size: 561
DSolve[y[x]*y'[x]==(a*x+b)^(-2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {a y(x) (a x+b) \operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryAi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryAiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}{a y(x) (a x+b) \operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\operatorname {AiryBi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \operatorname {AiryBiPrime}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}+c_1=0,y(x)\right ] \]