61.2.77 problem 77
Internal
problem
ID
[12083]
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
:
77
Date
solved
:
Tuesday, January 28, 2025 at 06:59:41 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }&=\alpha \,x^{k} y^{2}+\beta \,x^{s} y-\alpha \,\lambda ^{2} x^{k}+\beta \lambda \,x^{s} \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 164
dsolve((a*x^n+b*x^m+c)*diff(y(x),x)=alpha*x^k*y(x)^2+beta*x^s*y(x)-alpha*lambda^2*x^k+beta*lambda*x^s,y(x), singsol=all)
\[
y = \frac {-\alpha \left (\int \frac {x^{k} {\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+x^{m} b +c}d x}}{a \,x^{n}+x^{m} b +c}d x \right ) \lambda -\lambda c_{1} -{\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+x^{m} b +c}d x}}{c_{1} +\alpha \left (\int \frac {x^{k} {\mathrm e}^{-\int \frac {2 x^{k} \alpha \lambda -x^{s} \beta }{a \,x^{n}+x^{m} b +c}d x}}{a \,x^{n}+x^{m} b +c}d x \right )}
\]
✓ Solution by Mathematica
Time used: 4.922 (sec). Leaf size: 389
DSolve[(a*x^n+b*x^m+c)*D[y[x],x]==\[Alpha]*x^k*y[x]^2+\[Beta]*x^s*y[x]-\[Alpha]*\[Lambda]^2*x^k+\[Beta]*\[Lambda]*x^s,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha y(x) K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) K[2]^k}{(k-s) \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])}-\frac {\exp \left (-\int _1^{K[2]}-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right ) \left (-\alpha \lambda K[2]^k+\alpha K[3] K[2]^k+\beta K[2]^s\right )}{(k-s) \alpha \beta \left (b K[2]^m+a K[2]^n+c\right ) (\lambda +K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x-\frac {\beta K[1]^s-2 \alpha \lambda K[1]^k}{b K[1]^m+a K[1]^n+c}dK[1]\right )}{(k-s) \alpha \beta (\lambda +K[3])^2}\right )dK[3]=c_1,y(x)\right ]
\]