Internal problem ID [10507]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (a \ln \left (x \right )+b \right ) y^{\prime }-y^{2}-c \ln \left (x \right )^{n} y=-\lambda ^{2}+\lambda c \ln \left (x \right )^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 160
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)
\[ y \left (x \right ) = -\frac {\left (\left (\int \frac {{\mathrm e}^{\int \frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x \right ) {\mathrm e}^{\int -\frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x} \lambda +c_{1} {\mathrm e}^{\int -\frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x} \lambda +1\right ) {\mathrm e}^{\int \frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x}}{c_{1} +\int \frac {{\mathrm e}^{\int \frac {\ln \left (x \right )^{n} c -2 \lambda }{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x} \]
✓ Solution by Mathematica
Time used: 5.348 (sec). Leaf size: 275
DSolve[(a*Log[x]+b)*y'[x]==y[x]^2+c*(Log[x])^n*y[x]-\[Lambda]^2+\[Lambda]*c*(Log[x])^n,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}\frac {2 \lambda -c \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \left (c \log ^n(K[2])-\lambda +y(x)\right )}{c n (b+a \log (K[2])) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x\frac {2 \lambda -c \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right )}{c n (\lambda +K[3])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\frac {2 \lambda -c \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \left (c \log ^n(K[2])-\lambda +K[3]\right )}{c n (\lambda +K[3])^2 (b+a \log (K[2]))}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 \lambda -c \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right )}{c n (\lambda +K[3]) (b+a \log (K[2]))}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]