61.8.13 problem 22

Internal problem ID [12173]
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 : 22
Date solved : Tuesday, January 28, 2025 at 07:24:49 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 107

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

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

✓ Solution by Mathematica

Time used: 1.662 (sec). Leaf size: 275

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

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