61.4.6 problem 27
Internal
problem
ID
[12111]
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
:
27
Date
solved
:
Tuesday, January 28, 2025 at 12:24:54 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} y^{2}-a b \,x^{n} {\mathrm e}^{\lambda x} y+b n \,x^{n -1} \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 88
dsolve(diff(y(x),x)=a*exp(lambda*x)*y(x)^2-a*b*x^(n)*exp(lambda*x)*y(x)+b*n*x^(n-1),y(x), singsol=all)
\[
y = \frac {x^{n} \lambda \left (\int {\mathrm e}^{\lambda x +a b \left (\int x^{n} {\mathrm e}^{\lambda x}d x \right )}d x \right ) c_{1} a b +x^{n} a b -c_{1} \lambda \,{\mathrm e}^{a b \left (\int x^{n} {\mathrm e}^{\lambda x}d x \right )}}{a \left (\lambda \left (\int {\mathrm e}^{\lambda x +a b \left (\int x^{n} {\mathrm e}^{\lambda x}d x \right )}d x \right ) c_{1} +1\right )}
\]
✓ Solution by Mathematica
Time used: 3.332 (sec). Leaf size: 260
DSolve[D[y[x],x]==a*Exp[\[Lambda]*x]*y[x]^2-a*b*x^(n)*Exp[\[Lambda]*x]*y[x]+b*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {-a b c_1 \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a b \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )dK[2]+c_1 \lambda \exp \left (-\int _1^{e^{x \lambda }}-\frac {a b \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )-a b \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n}{a+a c_1 \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a b \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )dK[2]} \\
y(x)\to b \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^n-\frac {\lambda \exp \left (-\int _1^{e^{x \lambda }}-\frac {a b \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )}{a \int _1^{e^{x \lambda }}\exp \left (-\int _1^{K[2]}-\frac {a b \left (\frac {\log (K[1])}{\lambda }\right )^n}{\lambda }dK[1]\right )dK[2]} \\
\end{align*}