Internal problem ID [9989]
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: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-a \,{\mathrm e}^{\lambda x} y-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 83
dsolve(y(x)*diff(y(x),x)=a*exp(lambda*x)*y(x)+1,y(x), singsol=all)
\[ c_{1}-a \erf \left (\frac {\left (-y \relax (x ) \lambda +{\mathrm e}^{\lambda x} a \right ) \sqrt {2}}{2 \sqrt {-\lambda }}\right ) \sqrt {2}\, \sqrt {\pi }-2 \,{\mathrm e}^{\frac {a^{2} {\mathrm e}^{2 \lambda x}-2 \,{\mathrm e}^{\lambda x} a \lambda y \relax (x )-2 \left (-\frac {y \relax (x )^{2}}{2}+x \right ) \lambda ^{2}}{2 \lambda }} \sqrt {-\lambda } = 0 \]
✓ Solution by Mathematica
Time used: 1.008 (sec). Leaf size: 83
DSolve[y[x]*y'[x]==a*Exp[\[Lambda]*x]*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {a e^{\lambda x}}{\sqrt {\lambda }}=\frac {2 e^{\frac {\left (a e^{\lambda x}-\lambda y(x)\right )^2}{2 \lambda }}}{\sqrt {2 \pi } \text {Erfi}\left (\frac {\lambda y(x)-a e^{\lambda x}}{\sqrt {2} \sqrt {\lambda }}\right )+2 c_1},y(x)\right ] \]