Internal problem ID [10875]
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: 41.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (b \,x^{3} a -a \,x^{2}+b^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 74
dsolve(diff(y(x),x$2)+(a*x^3+2*b)*diff(y(x),x)+(a*b*x^3-a*x^2+b^2)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x \,{\mathrm e}^{-b x}+\frac {c_{2} {\mathrm e}^{-\frac {x \left (a \,x^{3}+8 b \right )}{8}} \left (x^{8} a^{2} \operatorname {WhittakerM}\left (\frac {3}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right )+7 \operatorname {WhittakerM}\left (\frac {11}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right ) a \,x^{4}+21 \operatorname {WhittakerM}\left (\frac {11}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right )\right )}{x^{\frac {11}{2}}} \]
✓ Solution by Mathematica
Time used: 0.431 (sec). Leaf size: 51
DSolve[y''[x]+(a*x^3+2*b)*y'[x]+(a*b*x^3-a*x^2+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {1}{8} e^{-b x} \left (8 c_1 x-\sqrt {2} c_2 \sqrt [4]{a x^4} \Gamma \left (-\frac {1}{4},\frac {a x^4}{4}\right )\right ) \]