Internal problem ID [9984]
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]]
Solve \begin {gather*} \boxed {y y^{\prime }-\frac {y}{\left (a x +b \right )^{2}}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 415
dsolve(y(x)*diff(y(x),x)=(a*x+b)^(-2)*y(x)+1,y(x), singsol=all)
\[ c_{1}+\frac {-a 2^{\frac {1}{3}} \left (x y \relax (x ) a^{2}+a y \relax (x ) b +1\right ) \AiryAi \left (-\frac {\left (-\frac {1}{2}+x^{2} \left (-\frac {y \relax (x )^{2}}{2}+x \right ) a^{4}+3 b \left (-\frac {y \relax (x )^{2}}{3}+x \right ) x \,a^{3}+\left (\left (-\frac {y \relax (x )^{2}}{2}+3 x \right ) b^{2}-x y \relax (x )\right ) a^{2}+b \left (b^{2}-y \relax (x )\right ) a \right ) 2^{\frac {2}{3}}}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (x a +b \right )^{2}}\right )-2 \AiryAi \left (1, -\frac {\left (-\frac {1}{2}+x^{2} \left (-\frac {y \relax (x )^{2}}{2}+x \right ) a^{4}+3 b \left (-\frac {y \relax (x )^{2}}{3}+x \right ) x \,a^{3}+\left (\left (-\frac {y \relax (x )^{2}}{2}+3 x \right ) b^{2}-x y \relax (x )\right ) a^{2}+b \left (b^{2}-y \relax (x )\right ) a \right ) 2^{\frac {2}{3}}}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (x a +b \right )^{2}}\right ) \left (a^{2}\right )^{\frac {2}{3}} \left (x a +b \right )}{a 2^{\frac {1}{3}} \left (x y \relax (x ) a^{2}+a y \relax (x ) b +1\right ) \AiryBi \left (-\frac {\left (-\frac {1}{2}+x^{2} \left (-\frac {y \relax (x )^{2}}{2}+x \right ) a^{4}+3 b \left (-\frac {y \relax (x )^{2}}{3}+x \right ) x \,a^{3}+\left (\left (-\frac {y \relax (x )^{2}}{2}+3 x \right ) b^{2}-x y \relax (x )\right ) a^{2}+b \left (b^{2}-y \relax (x )\right ) a \right ) 2^{\frac {2}{3}}}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (x a +b \right )^{2}}\right )+2 \AiryBi \left (1, -\frac {\left (-\frac {1}{2}+x^{2} \left (-\frac {y \relax (x )^{2}}{2}+x \right ) a^{4}+3 b \left (-\frac {y \relax (x )^{2}}{3}+x \right ) x \,a^{3}+\left (\left (-\frac {y \relax (x )^{2}}{2}+3 x \right ) b^{2}-x y \relax (x )\right ) a^{2}+b \left (b^{2}-y \relax (x )\right ) a \right ) 2^{\frac {2}{3}}}{2 \left (a^{2}\right )^{\frac {1}{3}} \left (x a +b \right )^{2}}\right ) \left (a^{2}\right )^{\frac {2}{3}} \left (x a +b \right )} = 0 \]
✓ Solution by Mathematica
Time used: 1.376 (sec). Leaf size: 561
DSolve[y[x]*y'[x]==(a*x+b)^(-2)*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {a y(x) (a x+b) \text {Ai}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\text {Ai}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \text {Ai}'\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}{a y(x) (a x+b) \text {Bi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+\text {Bi}\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )+2^{2/3} \sqrt [3]{a (a x+b)^3} \text {Bi}'\left (\frac {-2 x^3 a^4-6 b x^2 a^3+(b+a x)^2 y(x)^2 a^2-6 b^2 x a^2-2 b^3 a+2 (b+a x) y(x) a+1}{2 \sqrt [3]{2} \left (a (b+a x)^3\right )^{2/3}}\right )}+c_1=0,y(x)\right ] \]