61.8.4 problem 13
Internal
problem
ID
[12164]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.5-2
Problem
number
:
13
Date
solved
:
Tuesday, January 28, 2025 at 12:46:03 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=-\left (n +1\right ) x^{n} y^{2}+a \,x^{n +1} \ln \left (x \right )^{m} y-a \ln \left (x \right )^{m} \end{align*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 184
dsolve(diff(y(x),x)=-(n+1)*x^n*y(x)^2+a*x^(n+1)*(ln(x))^m*y(x)-a*(ln(x))^m,y(x), singsol=all)
\[
y = \frac {x^{-n -1} \left (x^{n +1} {\mathrm e}^{\int \frac {a \,x^{n +1} \ln \left (x \right )^{m} x -2 n -2}{x}d x}+\left (\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x \right ) n +\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x -c_{1} \right )}{\left (\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x \right ) n +\int x^{n} {\mathrm e}^{a \left (\int x^{n +1} \ln \left (x \right )^{m}d x \right )-2 \left (\int \frac {1}{x}d x \right ) \left (n +1\right )}d x -c_{1}}
\]
✓ Solution by Mathematica
Time used: 2.689 (sec). Leaf size: 248
DSolve[D[y[x],x]==-(n+1)*x^n*y[x]^2+a*x^(n+1)*(Log[x])^m*y[x]-a*(Log[x])^m,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {x^{-n-1} \left (c_1 x \exp \left (\int _1^x-\frac {-a K[1]^{n+2} \log ^m(K[1])+n+2}{K[1]}dK[1]\right )+c_1 (n+1) \int _1^x\exp \left (\int _1^{K[2]}-\frac {-a K[1]^{n+2} \log ^m(K[1])+n+2}{K[1]}dK[1]\right )dK[2]+n+1\right )}{(n+1) \left (1+c_1 \int _1^x\exp \left (\int _1^{K[2]}-\frac {-a K[1]^{n+2} \log ^m(K[1])+n+2}{K[1]}dK[1]\right )dK[2]\right )} \\
y(x)\to \frac {x^{-n} \left (\frac {\exp \left (\int _1^x-\frac {-a K[1]^{n+2} \log ^m(K[1])+n+2}{K[1]}dK[1]\right )}{\int _1^x\exp \left (\int _1^{K[2]}-\frac {-a K[1]^{n+2} \log ^m(K[1])+n+2}{K[1]}dK[1]\right )dK[2]}+\frac {n+1}{x}\right )}{n+1} \\
\end{align*}