Internal problem ID [10878]
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: 44.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 788
dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)=0,y(x), singsol=all)
\[ y = c_{1} +\left (\frac {\left (n +1\right ) \left (a x \,n^{2} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (-\frac {n}{2 \left (n +1\right )}, \frac {3}{2 \left (n +1\right )}+\frac {n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+2 a n x \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (-\frac {n}{2 \left (n +1\right )}, \frac {3}{2 \left (n +1\right )}+\frac {n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+n^{3} x^{-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (-\frac {n}{2 \left (n +1\right )}, \frac {3}{2 \left (n +1\right )}+\frac {n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+a x \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (-\frac {n}{2 \left (n +1\right )}, \frac {3}{2 \left (n +1\right )}+\frac {n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+4 n^{2} x^{-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (-\frac {n}{2 \left (n +1\right )}, \frac {3}{2 \left (n +1\right )}+\frac {n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+5 n \,x^{-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (-\frac {n}{2 \left (n +1\right )}, \frac {3}{2 \left (n +1\right )}+\frac {n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+2 x^{-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (-\frac {n}{2 \left (n +1\right )}, \frac {3}{2 \left (n +1\right )}+\frac {n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )\right )}{\left (2+n \right ) \left (3+2 n \right ) a}+\frac {\left (n +1\right )^{2} x^{-n} \left (\frac {a}{n +1}\right )^{\frac {1}{n +1}} \left (2+n \right ) \left (\frac {a \,x^{n +1}}{n +1}\right )^{-\frac {2+n}{2 \left (n +1\right )}} {\mathrm e}^{-\frac {a \,x^{n +1}}{2 \left (n +1\right )}} \operatorname {WhittakerM}\left (\frac {2+n}{2+2 n}, \frac {3+2 n}{2+2 n}, \frac {a \,x^{n +1}}{n +1}\right )}{\left (3+2 n \right ) a}\right ) c_{2} \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 56
DSolve[y''[x]+a*x^n*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2-\frac {c_1 x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {a x^{n+1}}{n+1}\right )}{n+1} \]