61.8.14 problem 23
Internal
problem
ID
[12174]
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
:
23
Date
solved
:
Tuesday, January 28, 2025 at 07:25:28 PM
CAS
classification
:
[_Riccati]
\begin{align*} \left (a \ln \left (x \right )+b \right ) y^{\prime }&=\ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+\lambda c \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 124
dsolve((a*ln(x)+b)*diff(y(x),x)=(ln(x))^n*y(x)^2+c*y(x)-lambda^2*(ln(x))^n+c*lambda,y(x), singsol=all)
\[
y = \frac {-\lambda c_{1} -\left (\int \frac {\ln \left (x \right )^{n} {\mathrm e}^{-\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x \right ) \lambda -{\mathrm e}^{-\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x}}{c_{1} +\int \frac {\ln \left (x \right )^{n} {\mathrm e}^{-\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x}
\]
✓ Solution by Mathematica
Time used: 1.808 (sec). Leaf size: 286
DSolve[(a*Log[x]+b)*D[y[x],x]==(Log[x])^n*y[x]^2+c*y[x]-\[Lambda]^2*(Log[x])^n+c*\[Lambda],y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \left (-\lambda \log ^n(K[2])+y(x) \log ^n(K[2])+c\right )}{c n (b+a \log (K[2])) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \log ^n(K[2])}{c n (\lambda +K[3]) (b+a \log (K[2]))}-\frac {\exp \left (-\int _1^{K[2]}-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \left (-\lambda \log ^n(K[2])+K[3] \log ^n(K[2])+c\right )}{c n (\lambda +K[3])^2 (b+a \log (K[2]))}\right )dK[2]-\frac {\exp \left (-\int _1^x-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right )}{c n (\lambda +K[3])^2}\right )dK[3]=c_1,y(x)\right ]
\]