61.2.73 problem 73
Internal
problem
ID
[12079]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
1.2.2.
Equations
Containing
Power
Functions
Problem
number
:
73
Date
solved
:
Tuesday, January 28, 2025 at 12:14:33 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x \left (a \,x^{k}+b \right ) y^{\prime }&=\alpha \,x^{n} y^{2}+\left (\beta -a n \,x^{k}\right ) y+\gamma \,x^{-n} \end{align*}
✓ Solution by Maple
Time used: 0.061 (sec). Leaf size: 138
dsolve(x*(a*x^k+b)*diff(y(x),x)=alpha*x^n*y(x)^2+(beta-a*n*x^k)*y(x)+gamma*x^(-n),y(x), singsol=all)
\[
y = -\frac {x^{-n} \left (\tanh \left (\frac {\left (\left (-b n -\beta \right ) \ln \left (a \,x^{k}+b \right )+k \left (\left (b n +\beta \right ) \ln \left (x \right )+c_{1} b \right )\right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \alpha \gamma +\beta ^{2}\right )}}{2 b k \left (b n +\beta \right )^{2}}\right ) \sqrt {\left (b n +\beta \right )^{2} \left (n^{2} b^{2}+2 b \beta n -4 \alpha \gamma +\beta ^{2}\right )}+\left (b n +\beta \right )^{2}\right )}{2 \alpha \left (b n +\beta \right )}
\]
✓ Solution by Mathematica
Time used: 2.948 (sec). Leaf size: 663
DSolve[x*(a*x^k+b)*D[y[x],x]==\[Alpha]*x^n*y[x]^2+(\[Beta]-a*n*x^k)*y[x]+\[Gamma]*x^(-n),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {x^{-n} \left (b \left (n \left (-\exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )-c_1 n \exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )+\exp \left (-\frac {(b n+\beta ) \left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right )}{b k}\right ) \left (\left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+\beta \right ) \left (-\exp \left (\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )-c_1 \left (\beta -\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}\right ) \exp \left (\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {-4 \alpha \gamma +b^2 n^2+\beta ^2+2 b \beta n}{\alpha \gamma }}+b n+\beta \right )}{2 b k}\right )\right )\right )}{2 \alpha \left (\exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+b n+\beta \right )}{2 b k}\right )+c_1 \exp \left (-\frac {\left (\log \left (a x^k+b\right )+\log (b)-k \log (x)+\log (k)\right ) \left (-\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}+b n+\beta \right )}{2 b k}\right )\right )} \\
y(x)\to \frac {x^{-n} \left (\sqrt {\alpha } \sqrt {\gamma } \sqrt {\frac {(b n+\beta )^2}{\alpha \gamma }-4}-b n-\beta \right )}{2 \alpha } \\
\end{align*}