Internal problem ID [10066]
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: 72.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y y^{\prime }+a \left (2 b x +1\right ) {\mathrm e}^{b x} y+a^{2} b \,x^{2} {\mathrm e}^{2 b x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 53
dsolve(y(x)*diff(y(x),x)+a*(1+2*b*x)*exp(b*x)*y(x)=-a^2*b*x^2*exp(2*b*x),y(x), singsol=all)
\[ y \relax (x ) = -\frac {{\mathrm e}^{b x} a \left (b x \RootOf \left (-{\mathrm e}^{\textit {\_Z}} b x -\expIntegral \left (1, -\textit {\_Z} \right )+c_{1}\right )-1\right )}{\RootOf \left (-{\mathrm e}^{\textit {\_Z}} b x -\expIntegral \left (1, -\textit {\_Z} \right )+c_{1}\right ) b} \]
✓ Solution by Mathematica
Time used: 0.477 (sec). Leaf size: 59
DSolve[y[x]*y'[x]+a*(1+2*b*x)*Exp[b*x]*y[x]==-a^2*b*x^2*Exp[2*b*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [b x e^{\frac {a e^{b x}}{a b x e^{b x}+b y(x)}}=\text {Ei}\left (\frac {a e^{b x}}{a b e^{b x} x+b y(x)}\right )+c_1,y(x)\right ] \]