61.8.7 problem 16

Internal problem ID [12167]
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 : 16
Date solved : Tuesday, January 28, 2025 at 07:24:44 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \ln \left (x \right )^{n} y^{2}+b \ln \left (x \right )^{m} y+b c \ln \left (x \right )^{m}-a \,c^{2} \ln \left (x \right )^{n} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 99

dsolve(diff(y(x),x)=a*(ln(x))^n*y(x)^2+b*(ln(x))^m*y(x)+b*c*(ln(x))^m-a*c^2*(ln(x))^n,y(x), singsol=all)
 

\[ y = \frac {-c a \left (\int \ln \left (x \right )^{n} {\mathrm e}^{-\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x}d x \right )-c_{1} c -{\mathrm e}^{-\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x}}{c_{1} +a \left (\int \ln \left (x \right )^{n} {\mathrm e}^{-\int \left (2 \ln \left (x \right )^{n} a c -\ln \left (x \right )^{m} b \right )d x}d x \right )} \]

✓ Solution by Mathematica

Time used: 1.448 (sec). Leaf size: 290

DSolve[D[y[x],x]==a*(Log[x])^n*y[x]^2+b*(Log[x])^m*y[x]+b*c*(Log[x])^m-a*c^2*(Log[x])^n,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right ) \left (-b \log ^m(K[2])+a c \log ^n(K[2])-a y(x) \log ^n(K[2])\right )}{a b (m-n) (c+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right )}{a b (m-n) (c+K[3])^2}-\int _1^x\left (-\frac {\exp \left (-\int _1^{K[2]}\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right ) \log ^n(K[2])}{b (m-n) (c+K[3])}-\frac {\exp \left (-\int _1^{K[2]}\left (2 a c \log ^n(K[1])-b \log ^m(K[1])\right )dK[1]\right ) \left (-b \log ^m(K[2])+a c \log ^n(K[2])-a K[3] \log ^n(K[2])\right )}{a b (m-n) (c+K[3])^2}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]