Internal problem ID [9742]
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: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_linear]
\[ \boxed {y^{\prime }-a \ln \left (x \right )^{n} y+a b x \ln \left (x \right )^{1+n} y-b \ln \left (x \right )-b=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 53
dsolve(diff(y(x),x)=a*(ln(x))^n*y(x)-a*b*x*(ln(x))^(n+1)*y(x)+b*ln(x)+b,y(x), singsol=all)
\[ y \left (x \right ) = \left (\int {\mathrm e}^{a \left (\int \ln \left (x \right )^{n} \left (-1+\ln \left (x \right ) b x \right )d x \right )} b \left (\ln \left (x \right )+1\right )d x +c_{1} \right ) {\mathrm e}^{\int \left (-\ln \left (x \right )^{1+n} a b x +a \ln \left (x \right )^{n}\right )d x} \]
✓ Solution by Mathematica
Time used: 0.535 (sec). Leaf size: 96
DSolve[y'[x]==a*(Log[x])^n*y[x]-a*b*x*(Log[x])^(n+1)*y[x]+b*Log[x]+b,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \exp \left (a \log ^{n+1}(x) (b \log (x) \operatorname {ExpIntegralE}(-n-1,-2 \log (x))-\operatorname {ExpIntegralE}(-n,-\log (x)))\right ) \left (\int _1^xb \exp \left (a \log ^{n+1}(K[1]) (\operatorname {ExpIntegralE}(-n,-\log (K[1]))-b \operatorname {ExpIntegralE}(-n-1,-2 \log (K[1])) \log (K[1]))\right ) (\log (K[1])+1)dK[1]+c_1\right ) \\ \end{align*}