61.2.32 problem 32
Internal
problem
ID
[12038]
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
:
32
Date
solved
:
Monday, January 27, 2025 at 11:52:51 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=-a n \,x^{n -1} y^{2}+c \,x^{m} \left (a \,x^{n}+b \right ) y-c \,x^{m} \end{align*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 199
dsolve(diff(y(x),x)=-a*n*x^(n-1)*y(x)^2+c*x^m*(a*x^n+b)*y(x)-c*x^m,y(x), singsol=all)
\[
y = \frac {a n \left (a \,x^{n}+b \right ) \left (\int \frac {x^{n -1} {\mathrm e}^{\frac {c \left (a \left (m +1\right ) x^{m +n +1}+b \,x^{m +1} \left (m +n +1\right )\right )}{\left (m +1\right ) \left (m +n +1\right )}}}{\left (a \,x^{n}+b \right )^{2}}d x \right )-x^{n} c_{1} a -c_{1} b +{\mathrm e}^{\frac {c \left (a \left (m +1\right ) x^{m +n +1}+b \,x^{m +1} \left (m +n +1\right )\right )}{\left (m +1\right ) \left (m +n +1\right )}}}{\left (a \left (\int \frac {x^{n -1} {\mathrm e}^{\frac {\left (a \left (m +1\right ) x^{n}+b \left (m +n +1\right )\right ) c \,x^{m} x}{\left (m +1\right ) \left (m +n +1\right )}}}{\left (a \,x^{n}+b \right )^{2}}d x \right ) n -c_{1} \right ) \left (a^{2} x^{2 n}+2 x^{n} a b +b^{2}\right )}
\]
✓ Solution by Mathematica
Time used: 6.524 (sec). Leaf size: 304
DSolve[D[y[x],x]==-a*n*x^(n-1)*y[x]^2+c*x^m*(a*x^n+b)*y[x]-c*x^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {a c_1 n \left (a x^n+b\right ) \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]+a^2 n x^n+c_1 e^{c x^{m+1} \left (\frac {a x^n}{m+n+1}+\frac {b}{m+1}\right )}+a b n}{a n \left (a x^n+b\right )^2 \left (1+c_1 \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]\right )} \\
y(x)\to \frac {\frac {e^{c x^{m+1} \left (\frac {a x^n}{m+n+1}+\frac {b}{m+1}\right )}}{a n \int _1^x\frac {\exp \left (c K[1]^{m+1} \left (\frac {a K[1]^n}{m+n+1}+\frac {b}{m+1}\right )\right ) K[1]^{n-1}}{\left (a K[1]^n+b\right )^2}dK[1]}+a x^n+b}{\left (a x^n+b\right )^2} \\
\end{align*}