Internal problem ID [10063]
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: 69.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y y^{\prime }-\left (a \,{\mathrm e}^{\lambda x}+b \right ) y-c \left (a^{2} {\mathrm e}^{2 \lambda x}+a b \left (\lambda x +1\right ) {\mathrm e}^{\lambda x}+b^{2} \lambda x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 551
dsolve(y(x)*diff(y(x),x)=(a*exp(lambda*x)+b)*y(x)+c*(a^2*exp(2*lambda*x)+a*b*(lambda*x+1)*exp(lambda*x)+b^2*lambda*x),y(x), singsol=all)
\[ -\frac {\left (3 c \lambda +1\right ) \left (6 \arctanh \left (\frac {6 b \,c^{2} \lambda ^{2} x +6 \,{\mathrm e}^{\lambda x} a \,c^{2} \lambda +2 b c \lambda x +2 \,{\mathrm e}^{\lambda x} a c +3 c \lambda y \relax (x )+y \relax (x )}{y \relax (x ) \sqrt {36 c^{3} \lambda ^{3}+33 c^{2} \lambda ^{2}+10 c \lambda +1}}\right ) c \lambda +2 \ln \left (\frac {9 \left (3 b c \,\lambda ^{2} x +3 \,{\mathrm e}^{\lambda x} a c \lambda +b \lambda x +{\mathrm e}^{\lambda x} a \right ) c}{y \relax (x )}\right ) \sqrt {36 c^{3} \lambda ^{3}+33 c^{2} \lambda ^{2}+10 c \lambda +1}-\ln \left (\frac {81 \left (9 b^{2} c^{3} \lambda ^{4} x^{2}+18 \,{\mathrm e}^{\lambda x} a b \,c^{3} \lambda ^{3} x +9 \,{\mathrm e}^{2 \lambda x} a^{2} c^{3} \lambda ^{2}+6 b^{2} c^{2} \lambda ^{3} x^{2}+12 \,{\mathrm e}^{\lambda x} a b \,c^{2} \lambda ^{2} x +9 b \,c^{2} \lambda ^{3} x y \relax (x )+6 \,{\mathrm e}^{2 \lambda x} a^{2} c^{2} \lambda +9 \,{\mathrm e}^{\lambda x} a \,c^{2} \lambda ^{2} y \relax (x )+b^{2} c \,\lambda ^{2} x^{2}-9 c^{2} \lambda ^{3} y \relax (x )^{2}+2 \,{\mathrm e}^{\lambda x} a b c \lambda x +6 b c \,\lambda ^{2} x y \relax (x )+{\mathrm e}^{2 \lambda x} a^{2} c +6 \,{\mathrm e}^{\lambda x} a c \lambda y \relax (x )-6 c \,\lambda ^{2} y \relax (x )^{2}+b \lambda x y \relax (x )+a \,{\mathrm e}^{\lambda x} y \relax (x )-\lambda y \relax (x )^{2}\right ) c}{\left (9 c \lambda +2\right ) y \relax (x )^{2}}\right ) \sqrt {36 c^{3} \lambda ^{3}+33 c^{2} \lambda ^{2}+10 c \lambda +1}+2 \arctanh \left (\frac {6 b \,c^{2} \lambda ^{2} x +6 \,{\mathrm e}^{\lambda x} a \,c^{2} \lambda +2 b c \lambda x +2 \,{\mathrm e}^{\lambda x} a c +3 c \lambda y \relax (x )+y \relax (x )}{y \relax (x ) \sqrt {36 c^{3} \lambda ^{3}+33 c^{2} \lambda ^{2}+10 c \lambda +1}}\right )\right )}{6 c \lambda \sqrt {36 c^{3} \lambda ^{3}+33 c^{2} \lambda ^{2}+10 c \lambda +1}}+\ln \left (b \lambda x +{\mathrm e}^{\lambda x} a \right )+\frac {\ln \left (b \lambda x +{\mathrm e}^{\lambda x} a \right )}{3 c \lambda }-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.306 (sec). Leaf size: 132
DSolve[y[x]*y'[x]==(a*Exp[\[Lambda]*x]+b)*y[x]+c*(a^2*Exp[2*\[Lambda]*x]+a*b*(\[Lambda]*x+1)*Exp[\[Lambda]*x]+b^2*\[Lambda]*x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {\frac {2 \text {ArcTan}\left (\frac {\frac {2 c \lambda y(x)}{a c e^{\lambda x}+b c \lambda x}-1}{\sqrt {-4 c \lambda -1}}\right )}{\sqrt {-4 c \lambda -1}}+\log \left (-\frac {c \lambda y(x)^2}{\left (a c e^{\lambda x}+b c \lambda x\right )^2}+\frac {y(x)}{a c e^{\lambda x}+b c \lambda x}+1\right )}{2 c \lambda }=\frac {\log \left (a e^{\lambda x}+b \lambda x\right )}{c \lambda }+c_1,y(x)\right ] \]