Internal problem ID [10885]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 51.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{n}+2 b \right ) y^{\prime }+\left (a b \,x^{n}-a \,x^{-1+n}+b^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 147
dsolve(diff(y(x),x$2)+(a*x^n+2*b)*diff(y(x),x)+(a*b*x^n-a*x^(n-1)+b^2)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x \,{\mathrm e}^{-b x}+c_{2} {\mathrm e}^{\frac {-a \,x^{n +1}-2 b x \left (n +1\right )}{2+2 n}} \left (\left (n +1\right ) \left (x^{-\frac {n}{2}} a +x^{-\frac {3 n}{2}-1} n \right ) \operatorname {WhittakerM}\left (\frac {-2-n}{2+2 n}, \frac {1+2 n}{2+2 n}, \frac {a \,x^{n +1}}{n +1}\right )+x^{-\frac {3 n}{2}-1} \operatorname {WhittakerM}\left (\frac {n}{2+2 n}, \frac {1+2 n}{2+2 n}, \frac {a \,x^{n +1}}{n +1}\right ) n^{2}\right ) \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+(a*x^n+2*b)*y'[x]+(a*b*x^n-a*x^(n-1)+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved