61.20.7 problem 40
Internal
problem
ID
[12309]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.8-2.
Equations
containing
arbitrary
functions
and
their
derivatives.
Problem
number
:
40
Date
solved
:
Tuesday, January 28, 2025 at 02:34:46 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=f^{\prime }\left (x \right ) y^{2}+a \,{\mathrm e}^{\lambda x} f \left (x \right ) y+a \,{\mathrm e}^{\lambda x} \end{align*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 117
dsolve(diff(y(x),x)=diff(f(x),x)*y(x)^2+a*exp(lambda*x)*f(x)*y(x)+a*exp(lambda*x),y(x), singsol=all)
\[
y = \frac {-f \,{\mathrm e}^{-\int \frac {-f^{2} {\mathrm e}^{\lambda x} a +2 f^{\prime }}{f}d x}-\int f^{\prime } {\mathrm e}^{a \left (\int f \,{\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {f^{\prime }}{f}d x \right )}d x -c_{1}}{f \left (c_{1} +\int f^{\prime } {\mathrm e}^{a \left (\int f \,{\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {f^{\prime }}{f}d x \right )}d x \right )}
\]
✓ Solution by Mathematica
Time used: 43.857 (sec). Leaf size: 319
DSolve[D[y[x],x]==D[ f[x],x]*y[x]^2+a*Exp[\[Lambda]*x]*f[x]*y[x]+a*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {a \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]\right )}{a f\left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right ) \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]\right )-c_1 \lambda } \\
y(x)\to \frac {a \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]}{\lambda -a f\left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right ) \exp \left (\int _1^{e^{x \lambda }}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a f\left (\frac {\log (K[1])}{\lambda }\right )}{\lambda }dK[1]\right )dK[2]} \\
\end{align*}