Internal problem ID [11090]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number: 256.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{n} a +b \right )^{2} y^{\prime \prime }+\left (x^{n} a +b \right ) \left (c \,x^{n}+d \right ) y^{\prime }+n \left (-d a +b c \right ) x^{-1+n} y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 82
dsolve((a*x^n+b)^2*diff(y(x),x$2)+(a*x^n+b)*(c*x^n+d)*diff(y(x),x)+n*(b*c-a*d)*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{\int \frac {-x^{n} c -d}{a \,x^{n}+b}d x}+c_{2} \left (\int {\mathrm e}^{-\left (\int \frac {-x^{n} c -d}{a \,x^{n}+b}d x \right )}d x \right ) {\mathrm e}^{\int \frac {-x^{n} c -d}{a \,x^{n}+b}d x} \]
✓ Solution by Mathematica
Time used: 0.923 (sec). Leaf size: 106
DSolve[(a*x^n+b)^2*y''[x]+(a*x^n+b)*(c*x^n+d)*y'[x]+n*(b*c-a*d)*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \exp \left (-\frac {x \left ((a d-b c) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {a x^n}{b}\right )+b c\right )}{a b}\right ) \left (\int _1^x\exp \left (\frac {\left (b c+(a d-b c) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {a K[1]^n}{b}\right )\right ) K[1]}{a b}\right ) c_1dK[1]+c_2\right ) \]