61.4.10 problem 31
Internal
problem
ID
[12115]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.3-2.
Equations
with
power
and
exponential
functions
Problem
number
:
31
Date
solved
:
Tuesday, January 28, 2025 at 12:25:14 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=-\left (k +1\right ) x^{k} y^{2}+a \,x^{k +1} {\mathrm e}^{\lambda x} y-a \,{\mathrm e}^{\lambda x} \end{align*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 184
dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+a*x^(k+1)*exp(lambda*x)*y(x)-a*exp(lambda*x),y(x), singsol=all)
\[
y = \frac {x^{-k -1} \left (x^{k +1} {\mathrm e}^{\int \frac {{\mathrm e}^{\lambda x} x^{k +1} a x -2 k -2}{x}d x}+\left (\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x \right ) k +\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x -c_{1} \right )}{\left (\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x \right ) k +\int x^{k} {\mathrm e}^{a \left (\int x^{k +1} {\mathrm e}^{\lambda x}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (k +1\right )}d x -c_{1}}
\]
✓ Solution by Mathematica
Time used: 46.167 (sec). Leaf size: 401
DSolve[D[y[x],x]==-(k+1)*x^k*y[x]^2+a*x^(k+1)*Exp[\[Lambda]*x]*y[x]-a*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {a \lambda \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]\right )}{a \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )+a c_1 \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]-c_1 \lambda ^2} \\
y(x)\to \frac {a \lambda \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]}{a \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\int _1^{e^{x \lambda }}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right ) \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a \left (\frac {\log (K[1])}{\lambda }\right )^{k+1}}{\lambda }dK[1]\right )dK[2]-\lambda ^2} \\
\end{align*}