61.24.69 problem 69
Internal
problem
ID
[12482]
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.
Problem
number
:
69
Date
solved
:
Tuesday, January 28, 2025 at 08:00:03 PM
CAS
classification
:
[[_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime } y&=\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 ) \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 257
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)
\[
\frac {\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, \left (\frac {c \lambda }{2}+\frac {1}{6}\right ) \ln \left (\frac {\left (3 c \lambda +1\right )^{2} \left (b^{2} c \,\lambda ^{2} x^{2}+2 \,{\mathrm e}^{\lambda x} a b c \lambda x +{\mathrm e}^{2 \lambda x} a^{2} c +b \lambda x y+a \,{\mathrm e}^{\lambda x} y-\lambda y^{2}\right ) c}{\left (9 c \lambda +2\right ) y^{2}}\right )-3 \left (c \lambda +\frac {1}{3}\right )^{2} \operatorname {arctanh}\left (\frac {\left (3 c \lambda +1\right ) \left (2 b c \lambda x +2 \,{\mathrm e}^{\lambda x} c a +y\right )}{\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, y}\right )+\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, \left (\left (-c \lambda -\frac {1}{3}\right ) \ln \left (\frac {\left (3 c \lambda +1\right ) \left (b \lambda x +{\mathrm e}^{\lambda x} a \right ) c}{y}\right )+\left (c \lambda +\frac {1}{3}\right ) \ln \left (b \lambda x +{\mathrm e}^{\lambda x} a \right )-c_{1} c \lambda \right )}{\sqrt {\left (4 c \lambda +1\right ) \left (3 c \lambda +1\right )^{2}}\, \lambda c} = 0
\]
✓ Solution by Mathematica
Time used: 0.163 (sec). Leaf size: 68
DSolve[y[x]*D[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],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{\frac {y(x)}{a e^{x \lambda } c+b x \lambda c}}\frac {1}{-c \lambda K[1]+1+\frac {1}{K[1]}}dK[1]=\frac {\log \left (a c e^{\lambda x}+b c \lambda x\right )}{c \lambda }+c_1,y(x)\right ]
\]